The linear stability of Reissner-Nordstr om spacetime for ...

The linear stability of Reissner-Nordstrom spacetime for small charge

Elena Giorgi

Submitted in partial fulfillment of therequirements for the degree of

Doctor of Philosophyin the Graduate School of Arts and Sciences

COLUMBIA UNIVERSITY

2019

Abstract

The linear stability of Reissner-Nordstrom spacetime for small charge

Elena Giorgi

In this thesis we prove the linear stability to gravitational and electromagnetic

perturbations of the Reissner-Nordstrom family of charged black holes with small

charge. Solutions to the linearized Einstein-Maxwell equations around a Reissner-

Nordstrom solution arising from regular initial data remain globally bounded on the

black hole exterior and in fact decay to a linearized Kerr-Newman metric. We express

the perturbations in geodesic outgoing null foliations, also known as Bondi gauge. To

obtain decay of the solution, one must add a residual pure gauge solution which is

proved to be itself controlled from initial data. Our results rely on decay statements

for the Teukolsky system of spin ˘2 and spin ˘1 satisfied by gauge-invariant null-

decomposed curvature components, obtained in earlier works. These decays are then

exploited to obtain polynomial decay for all the remaining components of curvature,

electromagnetic tensor and Ricci coefficients. In particular, the obtained decay is

optimal in the sense that it is the one which is expected to hold in the non-linear

problem.

Contents

Introduction 1

1 The Einstein-Maxwell equations in null frames 17

1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.1.1 Local null frames . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.1.2 S-tensor algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.2 Ricci coefficients, curvature and electromagnetic components . . . . . 22

1.2.1 Ricci coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.2.2 Curvature components . . . . . . . . . . . . . . . . . . . . . . 23

1.2.3 Electromagnetic components . . . . . . . . . . . . . . . . . . . 24

1.3 The Einstein-Maxwell equations . . . . . . . . . . . . . . . . . . . . . 25

1.3.1 Decomposition of Ricci and Riemann curvature . . . . . . . . 25

1.3.2 The null structure equations . . . . . . . . . . . . . . . . . . . 26

1.3.3 The Maxwell equations . . . . . . . . . . . . . . . . . . . . . . 30

1.3.4 The Bianchi equations . . . . . . . . . . . . . . . . . . . . . . 32

2 The Bondi gauge 37

2.1 Local Bondi gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.1.1 Local Bondi form of the metric . . . . . . . . . . . . . . . . . 38

i

2.1.2 Local normalized null frame . . . . . . . . . . . . . . . . . . . 38

2.2 Relations in the Bondi gauge . . . . . . . . . . . . . . . . . . . . . . . 39

2.3 Transport equations for average quantities . . . . . . . . . . . . . . . 42

3 Reissner-Nordstrom spacetime 44

3.1 Differential structure and metric . . . . . . . . . . . . . . . . . . . . . 45

3.1.1 Kruskal coordinate system . . . . . . . . . . . . . . . . . . . . 45

3.1.2 The Reissner-Nordstrom metric . . . . . . . . . . . . . . . . . 45

3.1.3 Double null coordinates u, v . . . . . . . . . . . . . . . . . . . 47

3.1.4 Standard coordinates t, r . . . . . . . . . . . . . . . . . . . . . 48

3.1.5 Ingoing Eddington-Finkelstein coordinates v, r . . . . . . . . 49

3.2 The Bondi form of the Reissner-Nordstrom metric . . . . . . . . . . . 49

3.2.1 Ricci coefficients and curvature components . . . . . . . . . . 51

3.3 Reissner-Nordstrom symmetries and operators . . . . . . . . . . . . . 52

3.3.1 Killing fields of the Reissner-Nordstrom metric . . . . . . . . . 52

3.3.2 The Su,r-tensor algebra in Reissner-Nordstrom . . . . . . . . . 53

3.3.3 Commutation formulae in Reissner-Nordstrom . . . . . . . . . 54

3.3.4 The l “ 0, 1 spherical harmonics . . . . . . . . . . . . . . . . . 55

4 The linearized gravitational and electromagnetic perturbations around

Reissner-Nordstrom 62

4.1 A guide to the formal derivation . . . . . . . . . . . . . . . . . . . . . 62

4.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.1.2 Outline of the linearization procedure . . . . . . . . . . . . . . 64

4.2 The full set of linearized equations . . . . . . . . . . . . . . . . . . . 72

4.2.1 The complete list of unknowns . . . . . . . . . . . . . . . . . . 73

4.2.2 Equations for the linearised metric components . . . . . . . . 73

ii

4.2.3 Linearized null structure equations . . . . . . . . . . . . . . . 74

4.2.4 Linearized Maxwell equations . . . . . . . . . . . . . . . . . . 76

4.2.5 Linearized Bianchi identities . . . . . . . . . . . . . . . . . . . 77

5 Special solutions: pure gauge and linearized Kerr-Newman 79

5.1 Pure gauge solutions G . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.1.1 Coordinate and null frame transformations . . . . . . . . . . . 80

5.1.2 Pure gauge solutions with jA “ 0 . . . . . . . . . . . . . . . . 86

5.1.3 Pure gauge solutions with g1 “ w1 “ w2 “ 0 . . . . . . . . . . 92

5.1.4 Gauge-invariant quantities . . . . . . . . . . . . . . . . . . . . 94

5.2 A 6-dimensional linearised Kerr-Newman family K . . . . . . . . . . 96

5.2.1 Linearized Kerr-Newman solutions with no angular momentum 96

5.2.2 Linearized Kerr-Newman solutions leaving the mass and the

charge unchanged . . . . . . . . . . . . . . . . . . . . . . . . . 99

6 The Teukolsky equations and the decay for the gauge-invariant quan-

tities 106

6.1 The spin ˘2 Teukolsky equations and the Regge-Wheeler system . . . 108

6.1.1 Generalized spin ˘2 Teukolsky system . . . . . . . . . . . . . 108

6.1.2 Generalized Regge-Wheeler system . . . . . . . . . . . . . . . 109

6.2 The spin ˘1 Teukolsky equation and the Fackerell-Ipser equation . . 111

6.2.1 Generalized spin ˘1 Teukolsky equation . . . . . . . . . . . . 111

6.2.2 Generalized Fackerell-Ipser equation in l “ 1 mode . . . . . . 112

6.3 The Chandrasekhar transformation . . . . . . . . . . . . . . . . . . . 113

6.4 Relation to the gravitational and electromagnetic perturbations of Reissner-

Nordstrom spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.4.1 Gravitational versus electromagnetic radiation . . . . . . . . . 116

iii

6.5 Boundedness and decay for the gauge-invariant quantities . . . . . . . 117

6.5.1 Proof of Theorem 6.5.1 . . . . . . . . . . . . . . . . . . . . . . 119

7 Initial data and well-posedness 131

7.1 Seed data on an initial cone . . . . . . . . . . . . . . . . . . . . . . . 131

7.2 Asymptotic flatness of initial data . . . . . . . . . . . . . . . . . . . . 132

7.3 The well-posedness theorem . . . . . . . . . . . . . . . . . . . . . . . 133

8 Gauge-normalized solutions and identification of the Kerr-Newman

parameters 138

8.1 The initial data normalization . . . . . . . . . . . . . . . . . . . . . . 139

8.2 The SU,R-normalization . . . . . . . . . . . . . . . . . . . . . . . . . . 141

8.3 Achieving the initial-data normalization for a general S . . . . . . . 144

8.4 Achieving the SU,R normalization for a bounded S . . . . . . . . . . 149

8.5 The Kerr-Newman parameters in l “ 0, 1 modes . . . . . . . . . . . . 158

9 Proof of boundedness 160

9.1 Initial data normalization and boundedness . . . . . . . . . . . . . . 160

9.1.1 The projection to the l “ 0 mode . . . . . . . . . . . . . . . . 164


9.1.3 The projection to the l ě 2 modes . . . . . . . . . . . . . . . . 176

9.1.4 The terms involved in the e3 direction . . . . . . . . . . . . . 182

9.1.5 The metric coefficients . . . . . . . . . . . . . . . . . . . . . . 183

9.2 Decay of the pure gauge solution GU,R . . . . . . . . . . . . . . . . . . 185

10 Proof of linear stability: decay 187

10.1 Statement of the theorem and outline of the proof . . . . . . . . . . . 187

10.2 Decay of the solution along the null hypersurface IU,R . . . . . . . . 192

iv


10.2.2 The projection to the l ě 2 modes . . . . . . . . . . . . . . . . 195



10.3 Decay of the solution S U,R in the exterior . . . . . . . . . . . . . . . 205

10.3.1 The projection to the l “ 1 mode: optimal decay in r and in u 207

10.3.2 The projection to the l ě 2 modes: optimal decay in r . . . . 211

10.3.3 The projection to the l ě 2 modes: optimal decay in u . . . . 212



10.3.6 Decay close to the horizon . . . . . . . . . . . . . . . . . . . . 223

Bibliography 225

Appendix A Explicit computations 231

A.1 Alternative expressions for qF and p . . . . . . . . . . . . . . . . . . . 231

A.2 Remarkable transport equations . . . . . . . . . . . . . . . . . . . . . 233

A.2.1 The charge aspect function . . . . . . . . . . . . . . . . . . . . 234

A.2.2 The mass-charge aspect function . . . . . . . . . . . . . . . . 237

Appendix B Proofs of Lemma 5.1.1.1 and Lemma 5.1.1.4 241

B.1 Proof of Lemma 5.1.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 241

B.2 Proof of Lemma 5.1.1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 246

v

Acknowledgements. The author would like to thank Sergiu Klainerman and Mu-

Tao Wang for their guidance and support. The author is also grateful to Jeremie

Szeftel, Pei-Ken Hung and Federico Pasqualotto for helpful discussions.

The author was supported by the Mathematics Department at Columbia Univer-

sity.

vi

Introduction

The problem of stability of the Kerr family pM, gM,aq in the context of the Ein-

stein vacuum equations occupies a center stage in mathematical General Relativity.

Roughly speaking, the problem of stability of the Kerr solution consists in showing

that all solutions of the Einstein vacuum equation

Ricpgq “ 0 (1)

which are spacetime developments of initial data sets sufficiently close to a member

of the Kerr family converge asymptotically to another member of the Kerr family.

The problem in the generality hereby formulated remains open, but many inter-

esting cases have been solved in the recent years. The only known proof of non-linear

stability with no symmetry assumption is the celebrated global stability of Minkowski

spacetime ([14]). A recent work proves the non-linear stability of Schwarzschild space-

time under a restrictive class of symmetry, which excludes rotating Kerr solutions as

final state of the evolution ([34]).

An important step to understand non-linear stability is proving linear stability,

which means proving boundedness and decay for the linearization of the Einstein

equation around the Kerr solutions. A first study of the linear stability of Schwarz-

schild spacetime to gravitational perturbations has been obtained in [16]. Different

1

results and proofs of the linear stability of the Schwarzschild spacetime have followed,

using the original Regge-Wheeler approach of metric perturbations (see [30]), and us-

ing wave gauge (see [31], [32], [33]). Steps towards the linear stabilty of Kerr solution

have been made in the proof of boundedness and decay for solutions to the Teukolsky

equations in Kerr in [36] and [17].

In this thesis we consider the above problems in the setting of Einstein-Maxwell

equations for charged black holes.

The problem of stability of charged black holes has as final goal the proof of

non-linear stability of Kerr-Newman family pM, gM,Q,aq as solutions to the Einstein-

Maxwell equation

Ricpgqµν “ T pF qµν :“ 2FµλFνλ´

1

2gµνF

αβFαβ (2)

where F is a 2-form satisfying the Maxwell equations

DrαFβγs “ 0, DαFαβ “ 0. (3)

The presence of a right hand side in the Einstein equation (2) and the Maxwell equa-

tions add new difficulties to the analysis of the problem, due to coupling between the

gravitational and the electromagnetic perturbations. This creates major difficulties

in both the analysis of the equations and the choice of the gauge, for which the en-

tanglement between the gravitational and the electromagnetic perturbation changes

the structure of the estimates and the choice of gauge.

An intermediate step towards the proof of non-linear stability of charged black

holes is the linear stability of the simplest non-trivial solution of the Einstein-Maxwell

equations, the Reissner-Nordstrom spacetime.

2

The Reissner-Nordstrom family of spacetimes pM, gM,Qq is most easily expressed

in local coordinates in the form:

gM,Q “ ´

ˆ

1´2M

r`Q2

r2

˙

dt2 `

ˆ

1´2M

r`Q2

r2

˙´1

dr2` r2

pdθ2` sin2 θdφ2

q, (4)

where M and Q are arbitrary parameters. The parameters M and Q may be inter-

preted as the mass and the charge of the source respectively. For physical reasons, it is

normally assumed that M ą |Q| (which excludes the case of naked singularity). This

spacetime reduces to Schwarzschild spacetime when Q “ 0 and the Kerr-Newman

metric gM,Q,a reduces to the Reissner-Nordstrom metric gM,Q for a “ 0.

The Reissner-Nordstrom spacetimes pM, gM,Qq are the simplest non-trivial solu-

tions to the Einstein-Maxwell equations and the unique electrovacuum spherically

symmetric spacetimes. It therefore plays for the Einstein-Maxwell equation the same

role as the Schwarzschild metric for the Einstein vacuum equation (1). It then makes

sense to start the study of the stability of charged black holes from the linearized

equations around Reissner-Nordstrom metric.

The purpose of the present thesis is to resolve the linear stability problem to

coupled gravitational and electromagnetic perturbations of the Reissner-Nordstrom

spacetime for small charge, i.e. the case of |Q| ! M . This is the first result on

quantitative stability of black holes coupled with matter, and is the electrovacuum

analogue of the linear stability of the Schwarzschild solution.

A first version of our main result can be stated as follows.

Theorem. (Linear stability of Reissner-Nordstrom: |Q| ! M) All solutions to the

linearized Einstein-Maxwell equations (in Bondi gauge) around Reissner-Nordstrom

with small charge arising from regular asymptotically flat initial data

1. remain uniformly bounded on the exterior and

3

2. decay according to a specific peeling1 to a standard linearised Kerr-Newman

solution

after adding a pure gauge solution which can itself be estimated by the size of the data.

The proof of linear stability roughly consists in two steps:

1. obtaining decay statements for gauge-invariant quantities,

2. choosing an appropriate gauge which allows to prove decay statements for the

gauge-dependent quantities.

In the first step, we need to identify the right gauge-invariant quantities which verify

wave equations which can be analyzed and for which quantitative decay statements

can be obtained. We completed the resolution of this part in our [27] and [28]. We

summarize the results in the following subsection.

The contribution of this thesis is the resolution of the second step. Once we obtain

decay for gauge-independent quantities from the first step, it is crucial to understand

the structure of the equations in order to choose just the right gauge conditions to

obtain decay for the gauge-dependent quantities. In particular, our goal here is to

obtain optimal decay for all quantities, where with optimal we mean decay which

would be consistent with bootstrap assumptions in the case of non-linear stability of

Reissner-Nordstrom spacetime.2 Having non-linear applications in mind, we aim to

obtain decay for all components, since they would all show up in the non-linear terms

of the wave equations.

In order to obtain the optimal decay for all components, we choose a particular

gauge ”far away” in time and space. This choice is inspired by the gauge choice in [34],

1The decay is consistent with the decay for the wave equation and with non-linear applications.2In particular, we obtain the same peeling decay of the bootstrap assumptions in the non-linear

stability of Schwarzschild in [34].

4

which allows for optimal decay for all components. We have to adapt this choice to

our case, where coupling between gravitational and electromagnetic radiation makes

the equations much more involved, and isolate quantities which transport decay would

be a difficult part of the problem. We summarize the main difficulties and the choice

of gauge in the last subsection of this introduction.

The gauge-invariant quantities and the Teukolsky

system

In the proof of linear stability of Schwarzschild in [16], the first step is the proof of

boundedness and decay for the solution of the spin ˘2 Teukolsky equation. These

are wave equations verified by the extreme null components of the curvature tensor

which decouple, to second order, from all other curvature components.

In linear theory, the Teukolsky equation, combined with cleverly chosen gauge

conditions, allows one to prove what is known as mode stability, i.e the lack of ex-

ponentially growing modes for all curvature components. Extensive literature by the

physics community covers these results (see for example [11], [13], [12] and [7]). This

weak version of stability is however far from sufficient to prove boundedness and de-

cay of the solution; one needs instead to derive sufficiently strong decay estimates to

hope to apply them in the nonlinear framework.

In [16], Dafermos, Holzegel and Rodnianski derive the first quantitative decay

estimates for the Teukolsky equations in Schwarzschild. The approach of [16] to

derive boundedness and quantitative decay for the Teukolsky equations relies on the

following ingredients:

1. A map which takes a solution to the Teukolsky equation, verified by the null

5

curvature component α, to a solution of a wave equation which is simpler to an-

alyze. In the case of Schwarzschild, this equation is known as the Regge-Wheeler

equation. The first such transformation was discovered by Chandrasekhar (see

[11]) in the context of mode decompositions and generalized by Wald in [45].

The physical version of this transformation first appears in [16].

2. A vectorfield-type method to get quantitative decay for the new wave equation.

3. A method by which we can derive estimates for solutions to the Teukolsky

equation from those of solutions to the transformed Regge -Wheeler equation.

Similarly, in the case of charged black holes, a key step towards the proof of

linear stability of Reissner-Nordstrom spacetime is to find an analogue of the Teukol-

sky equation and understand the behavior of their solution. The gauge-independent

quantities involved, analogous to α or α in vacuum, as well as the structure of the

equations that they verify, were identified in our earlier work [27]. We rely on the

following ingredients:

1. Computations in physical space which show the Teukolsky type equations ver-

ified by the extreme null curvature components in Reissner-Nordstrom space-

time. We obtain a system of two coupled Teukolsky-type equations.

2. A map which takes solutions to the above equations to solutions of a coupled

Regge-Wheeler-type equations.

3. A vectorfield method to get quantitative decay for the system. The analysis is

highly affected by the fact that we are dealing with a system, as opposed to a

single equation.

4. A method by which we can derive estimates for solutions to the Teukolsky-type

system from those of solutions to the transformed Regge-Wheeler-type system.

6

In [27], we derive the spin ˘2 Teukolsky-type system verified by two gauge-

independent curvature components of the gravitational and electromagnetic pertur-

bation of Reissner-Nordstrom. In addition to the Weyl curvature component α, we

introduce a new gauge-independent electromagnetic component f, which appears as

a coupling term to the Teukolsky-type equation for α. It is remarkable that such f

verifies itself a Teukolsky-type equation coupled back to α, as shown in [27]. The

quantities α and f verify a system of the schematic form:

$

’

’

&

’

’

%

lgM,Qα ` c1Lpαq ` c2Lpαq ` V1α “ Q ¨ Lpfq,

lgM,Qf` c1Lpfq ` c2Lpfq ` V2f “ ´Q ¨ Lpαq

(5)

where L and L are outgoing and ingoing null directions, and Q is the charge of the

spacetime. The presence of the first order terms Lpαq, Lpαq and Lpfq, Lpfq in (5)

prevents one from getting quantitative estimates to the system directly.

In order to derive appropriate decay estimates the system, new quantities q and qF

are defined, at the level of two and one derivative respectively of α and f. They cor-

respond to physical space versions of the Chandrasekhar transformations mentioned

earlier. This transformation has the remarkable property of turning the system of

Teukolsky type equations into a system of Regge-Wheeler-type equations. More pre-

cisely, it transforms the system (5) into the following schematic system:

$

’

’

&

’

’

%

lgM,Qq` V1q “ Q ¨Dď2qF,

lgM,QqF ` V2q

F “ Q ¨ q

(6)

where Dď2qF denotes a linear expression in terms of up to two derivative of qF. In the

case of zero charge, system (6) reduces to the first equation, i.e. the Regge-Wheeler

7

equation analyzed in [16].

In [27], we prove boundedness and decay for q and qF, and therefore for α and f.

We derive estimates for the system (6), by making use of the smallness of the charge

to absorb the right hand side through a combined estimate for the two equations.

Particularly problematic is the absorption in the trapping region, where the Morawetz

bulks in the estimates are degenerate. The specific structure of the terms appearing

in the system is exploited in order to obtain cancellation in this region.

Observe that the quantities pα, fq are symmetric-traceless two-tensors transporting

gravitational radiation, and therefore supported in l ě 2 spherical harmonics.

New feature in Reissner-Nordstrom: the projection

to l “ 1 spherical harmonics

In the linear stability of Schwarzschild spacetime to gravitational perturbations in [16],

the decay for α implies specific decay estimates for all the other curvature components

and Ricci coefficients supported in l ě 2 spherical harmonics, once a gauge condition

is chosen. In addition, an intermediate step of the proof is the following theorem:

Solutions of the linearized gravity around Schwarzschild supported only on l “ 0, 1

spherical harmonics are a linearized Kerr plus a pure gauge solution3.

In the setting of linear stability of Reissner-Nordstrom to coupled gravitational and

electromagnetic perturbations, we expect to have electromagnetic radiation supported

in l ě 1 spherical modes, as for solutions to the Maxwell equations in Schwarzschild

(see [9] or [39]).

On the other hand, the decay for the two tensors α and f obtained in [27] will

3Pure gauge solutions corresponds to coordinate transformations.

8

not give any decay information about the l “ 1 spherical mode of the perturbations.

It turns out that, in the case of solutions to the linearized gravitational and electro-

magnetic perturbations around Reissner-Nordstrom spacetime, the projection to the

l “ 0, 1 spherical harmonics is not exhausted by the linearized Kerr-Newman and

the pure gauge solutions. Indeed, the presence of the Maxwell equations involving

the extreme curvature component of the electromagnetic tensor, which is a one-form,

transports electromagnetic radiation supported in l ě 1 spherical harmonics. The

gauge-independent quantities involved in the electromagnetic radiation in Reissner-

Nordstrom were identified in our earlier work [28].

In [28], we have introduced a new gauge-independent one-form β, which is a

mixed curvature and electromagnetic component. This one-form has the additional

interesting property of vanishing for linearized Kerr-Newman solutions.

Such β verifies a spin ˘1 Teukolsky-type equation, with non-trivial right hand

side, which can be schematically written as

lgM,Q β ` c1Lpβq ` c2Lpβq ` V1β “ R.H.S. (7)

where the right hand side involves curvature components, electromagnetic compo-

nents and Ricci coefficients.

By applying the Chandrasekhar transformation, we obtain a derived quantity p at

the level of one derivative of β. Similar physical space versions of the Chandrasekhar

transformations were introduced [39]. This transformation has the remarkable prop-

erty of turning the Teukolsky-type equation (7) into a Fackerell-Ipser-type4 equation,

with right hand side which vanishes in l “ 1 spherical harmonics. Indeed, p verifies

4The Fackerell-Ipser equation was encountered in the study of Maxwell equations in Schwarzschildspacetime, see [39].

9

an equation of the schematic form:

lgM,Qp` V p “ Q ¨ div qF (8)

where the right hand side is supported in l ě 2 spherical harmonics.

Projecting equation (8) in l “ 1 spherical harmonics, we obtain a scalar wave

equation with vanishing right hand side, for which techniques developed in [22] and

[19] can be straightforwardly applied. This proves boundedness and decay for the

projection of p, and therefore β, to the l “ 1 spherical mode, in Reissner-Nordstrom

spacetimes with not-necessarily small charge.

The boundedness and decay for its projection into l ě 2 is implied by using the

result for the spin ˘2 Teukolsky equation in [27], for small charge.

The Main Theorems in [27] and [28] provide decay for the three quantities α, f,

β, and their negative spin equivalent α, f and β. Since these quantities are gauge-

independent, the above decay estimates do not depend on the choice of gauge.

The scope of this thesis is to show that the decays for these gauge-invariant quan-

tities imply boundedness and specific decay rates for all the remaining quantities

in the linear stability for coupled gravitational and electromagnetic perturbations of

Reissner-Nordstrom spacetime for small charge. The optimal decay for the gauge-

dependent quantities we are aiming to can be obtained only through a specific choice

of gauge. Such a choice of gauge is a crucial step, and will be discussed in the next

section.

10

Choice of gauge

In the linear stability of Schwarzschild [16], the perturbations of the metric are re-

stricted to the form of double null gauge. This choice still allows for residual gauge

freedom which in linear theory appears as the existence of pure gauge solutions.

Those are obtained from linearizing the families of metrics from applying coordinate

transformations which preserve the double null gauge of the metric.

In this work we use the Bondi gauge, inspired by the recent work on the non-linear

stability of Schwarzschild in [34]. In particular, we consider metric perturbations on

the outgoing null geodesic gauge, of the form

g “ ´2ςdudr ` ς2Ωdu2` gAB

ˆ

dθA ´1

2ςbAdu

˙ˆ

dθB ´1

2ςbBdu

˙

As in [16], this choice still allows for residual gauge freedom, corresponding to pure

gauge solutions.

The residual gauge freedom allows us to further impose gauge conditions which

are fundamental for the derivation of the specific decay rates of the gauge-dependent

quantities we want to achieve.

We make two choices of gauge-normalization: an initial-data normalization and a

far-away normalization. The motivation for the two choices of gauge-normalization

is different, and can be explained as follows.

The initial-data normalization consists of normalizing the solution on initial data

by adding an appropriate pure gauge solution which is explicitly computable from

the original solution’s initial data. This normalization allows to obtain boundedness

statements for the solution which is initial-data normalized, and also some good decay

statements for most components of the solution. Nevertheless, using this approach

11

there are components of the solution which do not decay in r, and this would be a

major obstacle in extending this result to the non-linear case. We call this type of

decay for the gauge-dependent components weak decay.

We make use of the weak decay derived through the initial-data normalized so-

lution to obtain boundedness in the whole exterior of the spacetime. Once we know

that the solution is bounded, we can define a normalization far-away which is the

correct one to obtain the optimal decay we want to achieve for each component of the

solution. This far-away normalization is inspired by the gauge choice done in [34].

More precisely, the normalization is realized by an ingoing null hypersurface for big r

and u. We should think of this null hypersurface as a bounded version of null infinity,

from which optimal decay for all the components can be derived in the past of it. We

call this type of decay for the gauge-dependent components strong decay.

By showing that those decays are independent of the chosen far-away position of

the null hypersurface, we obtain decay in the entire black hole exterior. In addition,

we can quantitatively control this new pure gauge solution in terms of the geometry

of initial data.

Decay of the gauge-dependent components

Using the initial-data normalization, we obtain by construction that some components

of the solution do not decay, or even grow, in r. More precisely, using initial-data

normalization we obtain for instance the following weak decay (see (9.64) for the

complete decay rates for all the components):

|ξ| ` |ω| ď Cu´1`δ, |b| ` |Ω| ď Cru´1`δ, |trγg| ď Cr2u´1`δ

12

The growth in r of these components is intrinsic to the initial-data normalization.

Indeed, the transport equation for ω (4.36) does not improve in powers of r in the

integration forward, so any integration forward starting from a bounded region of the

spacetime will not give any decay in r. Similarly for ξ.5

Strictly speaking, in linear theory this would not be an issue: it just proves a

weaker result. On the other hand, if we consider the linearization of the Einstein-

Maxwell equations as a first step towards the understanding of the non-linear stability

of black holes, we should obtain a decay which is consistent with bootstrap assump-

tions in the non-linear case. Having growth in r in some components will not allow to

close the analysis of the non-linear terms in the wave equations, and in the remaining

decay estimates.

In order to obtain the strong decay for all the components of our solution, we

define the normalization in the far-away hypersurface, inspired by the construction

of the ”last slice” in [34] and their choice of gauge. In all spherical harmonics, the

gauge is chosen so that the traces of the two null second fundamental forms vanish,

as in [34].

In addition, we define two new scalar functions, called charge aspect function and

mass-charge aspect function, respectively denoted ν (see (8.11)) and µ (see (8.12)),

which generalize the properties of the known mass-aspect function in the case of the

Einstein vacuum equation. Our generalization is essential to obtain the optimal decay

for all the components of the solution. These quantity are related to the Hawking

mass and the quasi local charge of the spacetime and verify good transport equations

5This issue is present also in the linear stability of Schwarzschild spacetime in [16], where thecomponent

(1)

ω does not decay in r for the same reason.

13

with integrable right hand sides:

Brpνl“1q “ 0

Brpµq “ Opr´1´δu´1`δq

In order to make use of these integrable transport equations, we impose these functions

to vanish along the ”last slice”.

The strong decay can be divided into an optimal decay in r (which would be

relevant in regions far-away in the spacetime) and an optimal decay in u (relevant in

regions far in the future). The optimal decay in r is easier to obtain, because there are

few transport equations which are integrable in r from far-away, if we only allow for

decay in u as u´12. For example, the transport equation for pχ (4.18) can be written

as:

Brpr2pχq “ ´r2α

Since α decays as r´3´δu´12`δ, we see that the right hand side is integrable in r. On

the other hand, α only decays as r´2´δu´1`δ, which would give a non-integrable right

hand side in the above transport equation.

To circumvent this difficulty, we identify a quantity Ξ (see (10.78)) which verifies

a transport equation with integrable right hand side:

BrpΞq “ Opr´1´δu´1`δq

The quantity Ξ is a combination of curvature, electromagnetic and Ricci coefficient

terms and generalizes a quantity (also denoted Ξ) in [34] which serves the same

purpose. Observe that, as opposed to the charge aspect function or the mass-charge

14

aspect function, Ξ decays fast enough along the ”last slice”, so that we do not need

to impose its vanishing along it.

Combining the decay of the above quantities we can prove that all the remaining

components verify the optimal decay in r and u which is consistent with non-linear

applications.6 More precisely, using the far-away normalization we obtain for instance

the following strong decay (see Theorem 10.1.1 for the complete decay rates for all

the components):

|ξ| ` |ω| ď Cr´1u´1`δ, |b| ` |Ω| ` |trγg| ď Cu´1`δ

Comparing with the above decay rate, we see that the far-away normalization sig-

nificantly improve the rate of decay and is the appropriate result to applications for

non-linear theory.

Outline of the thesis

We outline here the structure of the thesis.

In Chapter 1, we derive the general form of the Einstein-Maxwell equations written

with respect to a local null frame. In Chapter 2, we introduce our choice of gauge, the

Bondi gauge, to be used in the linear perturbations of Reissner-Nordstrom spacetime.

In Chapter 3, we describe the Reissner-Nordstrom spacetime, which is the solution

to the Einstein-Maxwell equations around which we perform the gravitational and

electromagnetic perturbations.

In Chapter 4, we derive the linearized Einstein-Maxwell equations around the

Reissner-Nordstrom solution. We denote a linear gravitational and electromagnetic

6In particular, we obtain the same decay as the bootstrap assumptions used in [34] in the caseof Schwarzschild.

15

perturbation of Reissner-Nordstrom a set of components which is a solution to those

equations.

In Chapter 5, we present special solutions to the linearized Einstein-Maxwell

equations around the Reissner-Nordstrom: pure gauge solutions and linearized Kerr-

Newman solutions.

In Chapter 6, we summarize the results on the boundedness and decay for the

solutions to the Teukolsky system as proved in [27] and [28]. We outline the procedure

to obtain such decay.

In Chapter 7, we present the characteristic initial problem and the well-posedness

of the linearized Einstein-Maxwell equations.

In Chapter 8 we describe the two gauge normalization which we will use and the

final Kerr-Newman parameters.

In Chapter 9, we prove boundedness of the solution using the initial data nor-

malization and in Chapter 10 we finally prove decay for all the gauge-dependent

components of the solution, therefore obtaining the proof of quantitative linear sta-

bility.

In Appendix A, we present explicit computations.

16

Chapter 1

The Einstein-Maxwell equations in

null frames

In this chapter, we derive the general form of the Einstein-Maxwell equations (2)

and (3) written with respect to a local null frame attached to a general foliation of

a Lorentzian manifold. In this chapter, we do not restrict to a specific form of the

metric and derive the main equations in their full generality. It is these equations

we shall linearize in Chapter 4 to obtain the equations for a linear gravitational and

electromagnetic perturbation of a spacetime.

We begin in Section 1.1 with preliminaries, recalling the notion of local null frame

and tensor algebra. In Section 1.2, we define Ricci coefficients, curvature and elec-

tromagnetic components of a solution to the Einstein-Maxwell equations. Finally, we

present the Einstein-Maxwell equations in Section 1.3.

17

1.1 Preliminaries

Let pM,gq be a 3 ` 1-dimensional Lorentzian manifold, and let D be the covariant

derivative associated to g.

1.1.1 Local null frames

Suppose that the the Lorentzian manifold pM,gq can be foliated by spacelike 2-

surfaces pS, gq, where g is the pullback of the metric g to S. To each point ofM, we

can associate a null frame N “ teA, e3, e4u, with teAuA“1,2 being tangent vectors to

pS, gq, such that the following relations hold:

g pe3, e3q “ 0, g pe4, e4q “ 0, g pe3, e4q “ ´2

g pe3, eAq “ 0 , g pe4, eAq “ 0 , g peA, eBq “ gAB .

(1.1)

The surfaces S will be identified in Chapter 2 as intersections of two specified hyper-

surfaces. Similarly, after a choice of gauge, the frame N can be identified explicitly

in terms of coordinates. See Section 2.1 for the identification of the null frame in the

Bondi gauge.

1.1.2 S-tensor algebra

In the following section, we will express the Ricci coefficients, curvature and elec-

tromagnetic components with respect to a null frame N associated to a foliation of

surfaces S. The objects we shall define are therefore S-tangent tensors. We recall

here the standard notations for operations on S-tangent tensors. (See [14] and [16])

18

S-projected Lie and covariant derivatives

We recall the definition of the projected covariant derivatives and the angular oper-

ator on S-tensors. We denote ∇ 3 “ ∇ e3 and ∇ 4 “ ∇ e4 the projection to S of the

spacetime covariant derivatives De3 and De4 respectively. We denote by Dχ and Dχ

the projected Lie derivative with respect to e3 and e4. The relations between them

are the following:

Df “ ∇ 4pfq,

DξA “ ∇ 4ξA ` χABξB,

DθAB “ ∇ 4θAB ` χACθCB ` χBCθA

C

(1.2)

and similarly for e3 replacing χ by χ.

Angular operators on S

We recall the following angular operators on S-tensors.

Let ξ be an arbitrary one-form and θ an arbitrary symmetric traceless 2-tensor

on S.

• ∇ denotes the covariant derivative associated to the metric g on S.

• D1 takes ξ into the pair of functions pdiv ξ, curl ξq, where

div ξ “ gAB∇ AξB, curl ξ “ εAB∇ AξB

• D‹1 is the formal L2-adjoint of D1, and takes any pair of functions pρ, σq into the

one-form ´∇ Aρ` εAB∇Bσ.

• D2 takes θ into the one-form D2θ “ pdiv θqC “ gAB∇ AθBC .

19

• D‹2 is the formal L2-adjoint of D2, and takes ξ into the symmetric traceless two

tensor

pD‹2ξqAB “ ´1

2

´

∇ BξA `∇ AξB ´ pdiv ξqgAB

¯

We can easily check that D‹k is the formal adjoint of Dk, i.e.

ż

S

pDkfqg “ż

S

fpD‹kgq

We recall the following L2 elliptic estimates. (see [14] or [34]).

Proposition 1.1.2.1. Let pS, gq be a compact surface with Gauss curvature K.

1. The following identity holds for a pair of function pρ, σq on S:

ż

S

`

|∇ ρ|2 ` |∇σ|2˘

“

ż

S

|D‹1pρ, σq|2 (1.3)

2. The following identity holds for 1-forms ξ on S:

ż

S

`

|∇ ξ|2 `K|ξ|2˘

“

ż

S

`

|div ξ|2 ` |curl ξ|2˘

“

ż

S

|D1ξ|2 (1.4)ż

S

`

|∇ ξ|2 ´K|ξ|2˘

“ 2

ż

S

|D‹2ξ|2 (1.5)

3. The following identity holds for symmetric traceless 2-tensors θ on S:

ż

S

`

|∇ θ|2 ` 2K|θ|2˘

“ 2

ż

S

|div θ|2 “ 2

ż

S

|D2θ|2 (1.6)

4. Suppose that the Gauss curvature is bounded. Then there exists a constant

C ą 0 such that the following estimate holds for all vectors ξ on S orthogonal

20

to the kernel of D‹2:

ż

S

1

r2|ξ|2 ď C

ż

S

|D‹2ξ|2 (1.7)

Given f a S-tensor, we define 4 f “ gAB∇ A∇ Bf . We recall the relations between

the angular operators and the laplacian 4 on S:

D1D‹1 “ ´4 0,

D‹1D1 “ ´4 1 `K,

D2D‹2 “ ´1

24 1 ´

1

2K,

D‹2D2 “ ´1

24 2 `K

(1.8)

where 4 0, 4 1 and 4 2 are the Laplacian on scalars, on 1-forms and on symmetric

traceless 2-tensors respectively, and K is the Gauss curvature of the surface S.

S-averages

Let f be a scalar on M. We define its S-average, and denote it by f as

f : “1

|S|

ż

S

f (1.9)

where |S| denotes the volume of pS, gq. We define the derived scalar function f as

qf :“ f ´ f. (1.10)

21

It follows from the definition that, for two functions f and g,

fg “ fg ` qfqg, (1.11)

|fg “ fg ´ fg “ qfg ` fqg ` p qfqg ´ qfqgq (1.12)

1.2 Ricci coefficients, curvature and electromag-

netic components

We now define the Ricci coefficients, curvature and electromagnetic components as-

sociated to the metric g with respect to the null frame N “ teA, e3, e4u, where the

indices A,B take values 1, 2. We follow the standard notations in [14].

1.2.1 Ricci coefficients

We define the Ricci coefficients associated to the metric g with respect to the null

frame N :

χAB : “ gpDAe4, eBq, χAB

:“ gpDAe3, eBq

ηA : “1

2gpD3e4, eAq, η

A:“

1

2gpD4e3, eAq,

ξA : “1

2gpD4e4, eAq, ξ

A:“

1

2gpD3e3, eAq

ω : “1

4gpD4e4, e3q, ω :“

1

4gpD3e3, e4q

ζA : “1

2gpDAe4, e3q,

(1.13)

We decompose the 2-tensor χAB into its tracefree part pχAB, a symmetric traceless

22

2-tensor on S, and its trace. We define

κ :“ trχ κ :“ trχ (1.14)

In particular we write χAB “12κ gAB ` pχAB, with gABpχAB “ 0 and κ “ gABχAB.

Similarly for χAB

.

It follows from (1.13) that we have the following relations for the covariant deriva-

tives of the null frame:

D4e4 “ ´2ωe4 ` 2ξAeA, D3e3 “ ´2ωe3 ` 2ξAeA,

D4e3 “ 2ωe3 ` 2ηAeA, D3e4 “ 2ωe4 ` 2ηAeA,

D4eA “ ηAe4 ` ξAe3, D3eA “ ηAe3 ` ξAe4,

DAe4 “ ´ζAe4 ` χABeB, DAe3 “ ζAe3 ` χABe

B,

DAeB “1

2χABe4 `

1

2χABe3.

(1.15)

The following relations for the commutators of the null frame also follow from (1.13):

“

e3, eA‰

“ pηA ´ ζAqe3 ` ξAe4 ´ χABeB,

“

e4, eA‰

“ pηA` ζAqe4 ` ξAe3 ´ χABe

B,

“

e3, e4

‰

“ ´2ωe3 ` 2ωe4 ` 2pηA ´ ηAqeA

(1.16)

1.2.2 Curvature components

Let W denote the Weyl curvature of g and let ‹W denote the Hodge dual on pM,gq

of W, defined by ‹Wαβγδ “12εαβµνW

µνγδ.

23

We define the null curvature components:

αAB : “ WpeA, e4, eB, e4q, αAB :“ WpeA, e3, eB, e3q

βA : “1

2WpeA, e4, e3, e4q, β

A:“

1

2WpeA, e3, e3, e4q

ρ : “1

4Wpe3, e4, e3, e4q σ :“

1

4‹Wpe3, e4, e3, e4q

(1.17)

The remaining components of the Weyl tensor are given by

WAB34 “ 2σεAB, WABC3 “ εAB‹βC, WABC4 “ ´εAB

‹βC ,

WA3B4 “ ´ρδAB ` σεAB, WABCD “ ´εABεCDρ

Observe that when interchanging e3 with e4, the one form β becomes ´β, the scalar

σ changes sign, while ρ remains unchanged.

1.2.3 Electromagnetic components

Let F be a 2-form in pM,gq, and let ‹F denote the Hodge dual on pM,gq of F,

defined by ‹Fαβ “12εµναβF

µν .

We define the null electromagnetic components:

pF qβA : “ FpeA, e4q,pF qβ

A:“ FpeA, e3q

pF qρ :“1

2Fpe3, e4q,

pF qσ :“1

2‹Fpe3, e4q

(1.18)

The only remaining component of F is given by FAB “ ´εABpF qσ.

Observe that when interchanging e3 with e4, the scalar pF qρ changes sign, while

pF qσ remains unchanged.

24

1.3 The Einstein-Maxwell equations

If pM,gq satisfies the Einstein-Maxwell equations

Rµν “ 2FµλFνλ´

1

2gµνF

αβFαβ, (1.19)

DrαFβγs “ 0, DαFαβ “ 0. (1.20)

the Ricci coefficients, curvature and electromagnetic components defined in (1.13),

(1.17) and (1.18) satisfy a system of equations, which is presented in this section.

1.3.1 Decomposition of Ricci and Riemann curvature

The Ricci curvature of pM,gq can be expressed in terms of the electromagnetic null

decomposition according to Einstein equation (1.19). We compute the following com-

ponents of the Ricci tensor.

RA3 “ 2FAλF3λ“ 2g

BCFABF3C ´ FA3F34 “ 2 pF qσεAC pF qβ

C´ 2 pF qρ pF qβ

A,

RA4 “ 2 pF qσεAC pF qβC ` 2 pF qρ pF qβA,

R33 “ 2gλµF3λF3µ “ 2gABF3AF3B “ 2 pF qβ ¨ pF qβ,

R34 “ g34pF34q

2` g

ABgCDFACFDB “ 2 pF qρ2

´ 2 pF qσ2

R44 “ 2 pF qβ ¨ pF qβ,

RAB “ 2FAλFBλ´ gABFD

λFλD `

1

2gABR34 “ ´p

pF qβpb pF qβqAB ` ppF qρ2

´pF qσ2

qgAB

We denote pb the symmetric traceless tensor product. Observe that as a consequence

of (1.19), the scalar curvature of the metric g is zero, therefore we have gACRAC “

R34.

25

Using the decomposition of the Riemann curvature in Weyl curvature and Ricci

tensor:

Rαβγδ “ Wαβγδ `1

2pgβδRαγ ` gαγRβδ ´ gβγRαδ ´ gαδRβγq, (1.21)

we can express the full Riemann tensor of pM,gq in terms of the above decomposi-

tions. We compute the following components of the Riemann tensor.

RA33B “ WA33B ´1

2gABR33 “ ´αAB ´

pF qβ ¨ pF qβgAB,

RA34B “ WA34B `RAB ´1

2gABR34 “ ρ gAB ´ p

pF qβpb pF qβqAB ´ σεAB,

RA334 “ WA334 ´RA3 “ 2βA´ 2 pF qσεA

C pF qβC` 2 pF qρ pF qβ

A,

R3434 “ W3434 ` 2R34 “ 4ρ` 4 pF qρ2´ 4 pF qσ2,

RA3CB “ WA3CB `1

2pgACR3B ´ gABR3Cq

“ εCB‹βA` gACp

pF qσεBC pF qβ

C´

pF qρ pF qβBq ´ gABp

pF qσεCD pF qβ

D´

pF qρ pF qβCq,

RABCD “ WABCD `1

2pgBDRAC ` gACRBD ´ gBCRAD ´ gADRBCq

We will use the above decompositions of Ricci and Riemann curvature in the deriva-

tion of the equations in Sections 1.3.2-1.3.4.

1.3.2 The null structure equations

The first equation for χ and χ is given by

∇ 3χAB ` χC

AχCB` 2ωχ

AB“ 2∇ BξA ` 2ηBξA ` 2η

AξB´ 4ζBξA `RA33B

∇ 4χAB ` χCAχCB ` 2ωχAB “ 2∇ BξA ` 2ηBξA ` 2ηAξB ` 4ζBξA `RA44B

26

We separate them in the symmetric traceless part, the trace part and the antisym-

metric part. We obtain respectively:

∇ 3pχ` κ pχ` 2ωpχ “ ´2D‹2ξ ´ α ` 2pη ` η ´ 2ζqpbξ

∇ 4pχ` κ pχ` 2ωpχ “ ´2D‹2ξ ´ α ` 2pη ` η ` 2ζqpbξ

(1.22)

∇ 3κ`1

2κ2` 2ωκ “ 2div ξ ´ pχ ¨ pχ` 2pη ` η ´ 2ζq ¨ ξ ´ 2 pF qβ ¨ pF qβ

∇ 4κ`1

2κ2` 2ωκ “ 2div ξ ´ pχ ¨ pχ` 2pη ` η ` 2ζq ¨ ξ ´ 2 pF qβ ¨ pF qβ

(1.23)

curl ξ “ ξ ^ pη ` η ´ 2ζq

curl ξ “ ξ ^ pη ` η ` 2ζq

(1.24)

The second equation for χ and χ is given by

∇ 4χAB “ 2∇ BηA ` 2ωχAB´ χCBχAC ` 2pξBξA ` ηBηAq `RA34B

∇ 3χAB “ 2∇ BηA ` 2ωχAB ´ χC

BχAC ` 2pξ

BξA ` ηBηAq `RA43B

We separate them in the symmetric traceless part, the trace part and the antisym-

metric part. We obtain respectively

∇ 3pχ`1

2κ pχ´ 2ωpχ “ ´2 D‹2η ´

1

2κpχ` ηpbη ` ξpbξ ´ pF qβpb pF qβ

∇ 4pχ`1

2κ pχ´ 2ωpχ “ ´2 D‹2η ´

1

2κpχ` ηpbη ` ξpbξ ´ pF qβpb pF qβ

(1.25)

27

∇ 3κ`1

2κκ´ 2ω κ “ 2div η ´ pχ ¨ pχ` 2ξ ¨ ξ ` 2η ¨ η ` 2ρ

∇ 4κ`1

2κκ´ 2ω κ “ 2div η ´ pχ ¨ pχ` 2ξ ¨ ξ ` 2η ¨ η ` 2ρ

(1.26)

curl η “ ´1

2χ^ χ` σ,

curl η “1

2χ^ χ´ σ

(1.27)

The equations for ζ are given by

∇ 3ζ “ ´2∇ω ´ χ ¨ pζ ` ηq ` 2ωpζ ´ ηq ` χ ¨ ξ ` 2ωξ ´1

2RA334,

´∇ 4ζ “ ´2∇ω ´ χ ¨ p´ζ ` ηq ` 2ωp´ζ ´ ηq ` χ ¨ ξ ` 2ωξ ´1

2RA443

and therefore reducing to

∇ 3ζ “ ´2∇ω ´ χ ¨ pζ ` ηq ` 2ωpζ ´ ηq ` χ ¨ ξ ` 2ωξ ´ β ` pF qσεAC pF qβ

C´

pF qρ pF qβ,

∇ 4ζ “ 2∇ω ` χ ¨ p´ζ ` ηq ` 2ωpζ ` ηq ´ χ ¨ ξ ´ 2ωξ ´ β ´ pF qσε ¨ pF qβ ´ pF qρ pF qβ

(1.28)

The equations for ξ and ξ are given by

∇ 4ξ ´∇ 3η “ 4ωξ ´ χ ¨ pη ´ ηq ´1

2RA334,

∇ 3ξ ´∇ 4η “ 4ωξ ` χ ¨ pη ´ ηq ´1

2RA443


∇ 4ξ ´∇ 3η “ ´χ ¨ pη ´ ηq ` 4ωξ ´ β ` pF qσε ¨ pF qβ ´ pF qρ pF qβ,

∇ 3ξ ´∇ 4η “ χ ¨ pη ´ ηq ` 4ωξ ` β ` pF qσε ¨ pF qβ ` pF qρ pF qβ

(1.29)

28

The equation for ω and ω is given by

∇ 4ω `∇ 3ω “ 4ωω ` ξ ¨ ξ ` ζ ¨ pη ´ ηq ´ η ¨ η `1

4R3434


∇ 4ω `∇ 3ω “ 4ωω ` ζ ¨ pη ´ ηq ` ξ ¨ ξ ´ η ¨ η ` ρ` pF qρ2´

pF qσ2 (1.30)

The spacetime equations that generate Codazzi equations are

∇ CχAB ` ζBχAC “ ∇ BχAC ` ζCχAB `RA3CB,

∇ CχAB ´ ζBχAC “ ∇ BχAC ´ ζCχAB `RA4CB

Taking the trace in C,A we obtain

div pχB“ ppχ ¨ ζqB ´

1

2κζB `

1

2p∇ Bκq ` βB `

pF qσεBC pF qβ

C´

pF qρ pF qβB,

div pχB “ ´ppχ ¨ ζqB `1

2κζB `

1

2p∇ Bκq ´ βB ` pF qσεB

C pF qβC `pF qρ pF qβB

(1.31)

The spacetime equation that generates Gauss equation is

g ACg BDRABCD “ 2K `1

2κκ´ pχ ¨ pχ

therefore reducing to

K “ ´1

4κκ`

1

2ppχ, pχq ´ ρ` pF qρ2

´pF qσ2 (1.32)

29

1.3.3 The Maxwell equations

For completeness, we derive here the null decompositions of Maxwell equations (1.20).

The equation DrαFβγs “ 0 gives three independent equations. The first one is

obtained in the following way:

0 “ DAF34 `D3F4A `D4FA3

“ ∇ AF34 ´ FpζAe3 ` χABeB, e4q ´ Fpe3,´ζAe4 ` χABe

Bq `∇ 3F4A

´Fp2ωe4 ` 2ηBeB, eAq

´Fpe4, ηAe3 ` ξAe4q `∇ 4FA3 ´ FpηAe4 ` ξAe3, e3q ´ FpeA, 2ωe3 ` 2ηBeBq

“ 2∇ A pF qρ´1

2κ pF qβA ´ ppχ ¨

pF qβqA `1

2κ pF qβ

A` ppχ ¨ pF qβqA ´∇ 3

pF qβA

`2ω pF qβA ´ 2ω pF qβA´ 2pηB ´ ηBqεAB

pF qσ ` 2pηA ` ηAqpF qρ`∇ 4

pF qβA

which reduces to

∇ 3pF qβA ´∇ 4

pF qβA“ ´

ˆ

1

2κ´ 2ω

˙

pF qβA `

ˆ

1

2κ´ 2ω

˙

pF qβA` 2∇ A pF qρ

` 2pηA ` ηAqpF qρ´ 2pηB ´ ηBqεAB

pF qσ ` ppχ ¨ pF qβqA ´ ppχ ¨pF qβqA

(1.33)

The second and third equation is obtained in the following way:

0 “ DAFB3 `DBF3A `D3FAB

“ ∇ AFB3 ´ FpeB, ζAe3 ` χACeCq `∇ BF3A ´ FpζBe3 ` χBCe

C , eAq `∇ 3FAB

´FpηAe3 ` ξAe4, eBq ´ FpeA, ηBe3 ` ξBe4q

“ ∇ A pF qβB ´∇ BpF qβ

A´ pζA ´ ηAq

pF qβB` pζB ´ ηBq

pF qβA` ξ

A

pF qβB ´ ξBpF qβA

`pχACεCB ` χBCε

CAq

pF qσ ´ εAB∇ 3pF qσ

30

Contracting with εAB we obtain

∇ 3pF qσ ` κ pF qσ “ curl pF qβ ´ pζ ´ ηq ^ pF qβ ` ξ ^ pF qβ,

∇ 4pF qσ ` κ pF qσ “ curl pF qβ ` pζ ` ηq ^ pF qβ ` ξ ^ pF qβ

(1.34)

The equation DµFµν “ gBCDBFCν ´12D4F3ν ´

12D3F4ν “ 0 gives three additional

independent equations. The first one is obtained in the following way:

0 “ gBCDBFCA ´

1

2D4F3A ´

1

2D3F4A

“ gBCp´εCA∇ B

pF qσ ´ Fp1

2χBCe4 `

1

2χBCe3, eAq ´ FpeC ,

1

2χABe4 `

1

2χABe3qq

`1

2∇ 4

pF qβA`

1

2Fp2ωe3 ` 2ηCeC , eAq `

1

2Fpe3, ηAe4 ` ξAe3q

`1

2∇ 3

pF qβA `1

2Fp2ωe4 ` 2ηceC , eAq `

1

2Fpe4, ηAe3 ` ξAe4q

“ εAC∇ C pF qσ `1

4κ pF qβA `

1

4κ pF qβ

A´

1

2ppχ ¨ pF qβqA ´

1

2ppχ ¨ pF qβqA ` p´ηA ` ηAq

pF qρ

`1

2∇ 4

pF qβA´ ω pF qβ

A´ ω pF qβA `

1

2∇ 3

pF qβA ` pηB` ηBqεAB

pF qσ

which reduces to

∇ 3pF qβA `∇ 4

pF qβA“ ´

ˆ

1

2κ´ 2ω

˙

pF qβA ´

ˆ

1

2κ´ 2ω

˙

pF qβA` 2pηA ´ ηAq

pF qρ

´ 2εAC∇ C pF qσ ´ 2pηB ` ηBqεABpF qσ ` ppχ ¨ pF qβqA ` ppχ ¨

pF qβqA

(1.35)

Summing and subtracting (1.33) and (1.35) we obtain

∇ 3pF qβA `

ˆ

1

2κ´ 2ω

˙

pF qβA “ ´D‹1p pF qρ, pF qσq ` 2ηApF qρ´ 2ηBεAB

pF qσ ` ppχ ¨ pF qβqA

∇ 4pF qβ

A`

ˆ

1

2κ´ 2ω

˙

pF qβA“ D‹1p pF qρ,´ pF qσq ´ 2η

A

pF qρ´ 2ηBεABpF qσ ` ppχ ¨ pF qβqA

(1.36)

31

The last two equations are given by

0 “ gBCDBFC4 ´

1

2D4F34

“ gBCp∇ B pF qβC ´ Fp

1

2χBCe4 `

1

2χBCe3, e4q ´ FpeC ,´ζBe4 ` χBAe

Aqq

´1

2p2∇ 4

pF qρ´ Fp2ωe3 ` 2ηAeA, e4q ´ Fpe3,´2ωe4 ` 2ξAeAqq

“ div pF qβ ´ pχABεAB pF qσ ´ κ pF qρ´∇ 4

pF qρ` ppζ ` ηq ¨ pF qβq ´ ξ ¨ pF qβ

which reduces to

∇ 4pF qρ` κ pF qρ “ div pF qβ ` pζ ` ηq ¨ pF qβ ´ ξ ¨ pF qβ,

∇ 3pF qρ` κ pF qρ “ ´div pF qβ ` pζ ´ ηq ¨ pF qβ ´ ξ ¨ pF qβ

(1.37)

1.3.4 The Bianchi equations

The Bianchi identities for the Weyl curvature are given by

DαWαβγδ “1

2pDγRβδ ´DδRβγq “: Jβγδ

DrσWγδsαβ “ gδβJαγσ ` gγαJβδσ ` gσβJαδγ ` gδαJβσγ ` gγβJασδ ` gσαJβγδ :“ Jσγδαβ

The Bianchi identities for α and α are given by

∇ 3αAB `1

2καAB ´ 4ωαAB “ ´2pD‹2 βqAB ´ 3ppχABρ`

‹pχABσq ` ppζ ` 4ηqbβqAB `

`1

2pJ3A4B4 ` J3B4A4 ` J434gABq

∇ 4αAB `1

2καAB ´ 4ωαAB “ 2pD‹2 βqAB ´ 3ppχ

ABρ` ‹

pχABσq ´ pp´ζ ` 4ηqbβqAB `

`1

2pJ4A3B3 ` J4B3A3 ` J343gABq

32

Using that J3A4B4 “ ´gABJ434 ` 2JBA4, it is reduced to

∇ 3αAB `1

2καAB ´ 4ωαAB “ ´2pD‹2 βqAB ´ 3ppχABρ`

‹pχABσq ` ppζ ` 4ηqbβqAB`

`JBA4 ` JAB4 ´1

2gABJ434,

∇ 4αAB `1

2καAB ´ 4ωαAB “ 2pD‹2 βqAB ´ 3ppχ

ABρ` ‹

pχABσq ´ pp´ζ ` 4ηqbβqAB`

`JBA3 ` JAB3 ´1

2gABJ343

(1.38)

The Bianchi identities for β and β are given by

∇ 4βA ` 2κβA ` 2ωβA “ div αA ` pp2ζ ` ηq ¨ αqA ` 3pξAρ`‹ξAσq ´ J4A4,

∇ 3βA ` 2κβA` 2ωβ

A“ ´div αA ` pp2ζ ´ ηq ¨ αqA ´ 3pξ

Aρ` ‹ξ

Aσq ` J3A3

(1.39)

and

∇ 3βA ` κβA ´ 2ω βA “ D‹1p´ρ, σqA ` 2ppχ ¨ βqA ` ξ ¨ α ` 3pηAρ`‹ηA σq ` J3A4,

∇ 4βA ` κβA ´ 2ω βA“ D‹1pρ, σqA ` 2ppχ ¨ βqA ´ ξ ¨ α ´ 3pη

Aρ´ ‹η

Aσq ´ J4A3

(1.40)

The Bianchi identity for ρ is given by

∇ 4ρ`3

2κρ “ div β ` p2η ` ζq ¨ β ´

1

2ppχ ¨ αq ´ 2ξ ¨ β ´

1

2J434,

∇ 3ρ`3

2κρ “ ´div β ´ p2η ´ ζq ¨ β `

1

2ppχ ¨ αq ` 2ξ ¨ β ´

1

2J343

(1.41)

The Bianchi identity for σ is given by

∇ 4σ `3

2κσ “ ´ curl β ´ p2η ` ζq ^ β `

1

2pχ^ α ´

1

2‹J434,

∇ 3σ `3

2κσ “ ´ curl β ´ p2η ´ ζq ^ β ´

1

2pχ^ α `

1

2‹J343

33

and writing ‹J434 “12J4µνε

µν34 “ ´J4ABεAB “ pJAB4 ´ JBA4qε

AB, we obtain

∇ 4σ `3

2κσ “ ´ curl β ´ p2η ` ζq ^ β `

1

2pχ^ α ´

1

2pJAB4 ´ JBA4qε

AB,

∇ 3σ `3

2κσ “ ´ curl β ´ p2η ´ ζq ^ β ´

1

2pχ^ α `

1

2pJAB3 ´ JBA3qε

AB

(1.42)

We compute the following Js, needed in the derivation of the above Bianchi iden-

tities.

2J434 “ D3R44 ´D4R43

“ ∇ 3pR44q ´ 2RpD3e4, e4q ´∇ 4pR34q `RpD4e4, e3q `Rpe4,D4e3q

“ 2∇ 3ppF qβ ¨ pF qβq ´∇ 4p2

pF qρ2´ 2 pF qσ2

q ´ 2Rp2ωe4 ` 2ηAeA, e4q

`Rp´2ωe4, e3q `Rpe4, 2ωe3 ` 2ηAeAq

“ 2∇ 3ppF qβ ¨ pF qβq ´ 2∇ 4p

pF qρ2´

pF qσ2q ´ 4ωR44 ` 4pηA ´ ηAqR4A

“ 2∇ 3ppF qβ ¨ pF qβq ´ 2∇ 4p

pF qρ2´

pF qσ2q ´ 8ωp pF qβ, pF qβq

´8pηA ´ ηAqp pF qσεAC pF qβC `

pF qρ pF qβAq

2J4A4 “ DAR44 ´D4R4A “ ∇ ApR44q ´ 2RpDAe4, e4q ´∇ 4pR4Aq

`RpD4e4, eAq `Rpe4,D4eAq

“ 2∇ Ap pF qβ ¨ pF qβq ´ 2∇ 4ppF qσεA

C pF qβC `pF qρ pF qβAq ´ 2Rp´ζAe4 ` χABe

B, e4q

`Rp´2ωe4, eAq `Rpe4, ηAe4q

“ 2∇ Ap pF qβ ¨ pF qβq ´ 2∇ 4ppF qσεA

C pF qβC `pF qρ pF qβAq

´4χABppF qσεAC pF qβC `

pF qρ pF qβBq

´4ωp pF qσεAC pF qβC `

pF qρ pF qβAq ` 2p2ζA ` ηAqppF qβ, pF qβq

34

2J3A4 “ DAR43 ´D4R3A

“ ∇ ApR34q ´RpDAe4, e3q ´Rpe4,DAe3q ´∇ 4pR3Aq

`RpD4e3, eAq `Rpe3,D4eAq

“ ∇ Ap2 pF qρ2´ 2 pF qσ2

q ´∇ 4p2pF qσεA

C pF qβC´ 2 pF qρ pF qβ

Aq `

´Rp´ζAe4 ` χABeB, e3q ´Rpe4, ζAe3 ` χABe

Bq `Rp2ωe3 ` 2ηBeB, eAq

`Rpe3, ηAe4q

“ 2∇ Ap pF qρ2´

pF qσ2q ´∇ 4p2

pF qσεAC pF qβ

C´ 2 pF qρ pF qβ

Aq

´χABp2pF qσεBC pF qβ

C´ 2 pF qρ pF qβBq

´χABp2 pF qσεBC pF qβC ` 2 pF qρ pF qβBq ` 2ωp2 pF qσεA

C pF qβC´ 2 pF qρ pF qβ

Aq

`2ηBp´p pF qβpb pF qβqAB ` ppF qρ2

´pF qσ2

qgABq ` ηAp2pF qρ2

´ 2 pF qσ2q

2JAB4 “ DBR4A ´D4RAB

“ ∇ BpR4Aq ´RpDBe4, eAq ´Rpe4,DBeAq ´∇ 4pRABq

`RpD4eA, eBq `RpeA,D4eBq

“ ∇ Bp2 pF qσεAC pF qβC ` 2 pF qρ pF qβAq

´∇ 4p´ppF qβpb pF qβqAB ` p

pF qρ2´

pF qσ2qgABq ´Rp´ζBe4 ` χBCe

C , eAq

´Rpe4,1

2χABe4 `

1

2χABe3q `Rpη

Ae4, eBq `RpeA, ηBe4q

“ ∇ Bp2 pF qσεAC pF qβC ` 2 pF qρ pF qβAq

´∇ 4p´ppF qβpb pF qβqAB ` p

pF qρ2´

pF qσ2qgABq `

`pζB ` ηBqp2pF qσεA

C pF qβC ` 2 pF qρ pF qβAq ` χBCppF qβpb pF qβqA

Cq

´2χABppF qρ2

´pF qσ2

q

´χABppF qβ, pF qβq ` η

Ap2 pF qσεB

C pF qβC ` 2 pF qρ pF qβBq

35

We shall make use of these computations, simplified using Maxwell equations, in

deriving the linearized Bianchi identities in Section 4.2.5.

36

Chapter 2

The Bondi gauge

In this chapter, we introduce the choice of gauge we use throughout the paper to

perform the perturbation of the solution to the Einstein-Maxwell equations. This

choice of gauge introduces a restriction on the form of the metric g on M, which

nevertheless does not saturate the gauge freedom of the Einstein-Maxwell equations.1

Our choice of gauge is the outgoing geodesic foliation, also called Bondi gauge.

This choice of coordinates is particularly suited to exploit properties of decay towards

null infinity, which we will take advantage of.

We begin in Section 2.1 with the definition of local Bondi gauge. In Section 2.2,

we derive the equations for the metric components and the Ricci coefficients implied

by the Bondi gauge. These equations will be added to the set of Einstein-Maxwell

equations derived in Section 1.3. In Section 2.3, we derive the equations for the

average quantities in a Bondi gauge, which are used later in the derivation of the

linearized equations for scalars.

1The gauge freedom remaining will be exploited later by the pure gauge solutions (see Section5.1)

37

2.1 Local Bondi gauge

Let pM,gq be a 3` 1 dimensional Lorentzian manifold.

2.1.1 Local Bondi form of the metric

In a neighborhood of any point p PM, we can introduce local coordinates pu, s, θ1, θ2q

such that the metric can be expressed in the following Bondi form (see [10]):

g “ ´2ςduds` ς2Ωdu2` gAB

ˆ

dθA ´1

2ςbAdu

˙ˆ

dθB ´1

2ςbBdu

˙

(2.1)

for two spacetime functions Ω, ς :MÑ R, with ς ‰ 0, a Su,s-tangent vector bA and a

Su,s-tangent covariant symmetric 2-tensor gAB. Here Su,s denotes the two-dimensional

Riemannian manifold (with metric g) obtained as intersection of the hypersurfaces of

constant u and s.

Note that tu “ constantu are outgoing null hypersurfaces for g.

2.1.2 Local normalized null frame

We define a normalized outgoing geodesic null frame N “ teA, e3, e4u associated to

the above coordinates as follows. We define

e3 “ 2ς´1Bu ` ΩBs ` b

ABθA , e4 “ Bs, eA “ BθA (2.2)

Observe that relations (1.1) hold. In particular, notice that the surfaces Su,s

define a foliation of the spacetime of the type described in Section 1.1.1, therefore

the decomposition in null frame of Ricci coefficients, curvature and electromagnetic

components described above can be applied to this case.

38

To the foliation Su,s we can associate a scalar function rpu, sq defined by

|Su,s| “ 4πrpu, sq2 (2.3)

where |Su,s| is the area of pSu,s, gq.

2.2 Relations in the Bondi gauge

The restriction to perturbations of the metric of the form (2.1) verifying the Einstein-

Maxwell equations gives additional relations between the Ricci coefficients as defined

in Section 1.2.1. We summarize them in the following lemma.

Lemma 2.2.0.1. The Ricci coefficients associated to a metric g of the form (2.1)

with respect to the null frame (2.2) verify:

ξA “ 0, ω “ 0, ηA“ ´ζA (2.4)

The metric components satisfy

∇ Aς “ ηA ´ ζA, (2.5)

∇ 4ς “ 0 (2.6)

∇ AΩ “ ´ξA` Ω pζA ´ ηAq , (2.7)

∇ 4Ω “ ´2ω, (2.8)

∇ 4bA ´ χABbB“ ´2 pηA ` ζAq , (2.9)

BspgABq “ 2pχAB ` κgAB, (2.10)

2ς´1BupgABq ` ΩBspgABq “ 2pχ

AB` 2pD‹2bqAB ` pκ´ div bqgAB (2.11)

39

Proof. The vectorfield Bs is geodesic, i.e. De4e4 “ 0. Using (1.15), this implies ω “ 0

and ξA “ 0.

Since e3puq “ 2ς´1 and e4puq “ eApuq “ 0, we can apply“

e3, eA‰

and re3, e4s to u

and using (1.16) we obtain

“

e3, eA‰

u “ ∇ 3∇ Apuq ´∇ Ap∇ 3puqq “ ´2∇ Apς´1q “ 2ς´2∇ Apςq

“

e3, eA‰

u “ pηA ´ ζAqe3puq ` ξAe4puq ´ χABeBpuq “ 2pηA ´ ζAqς

´1

“

e3, e4

‰

u “ e3e4puq ´ e4e3puq “ ´2∇ 4pς´1q “ 2ς´2∇ 4pςq

“

e3, e4

‰

u “ 2ωe4puq ` 2pηB ´ ηBqeBpuq “ 0

Since e3psq “ Ω, e4psq “ 1, eApsq “ 0, we can apply“

e4, eA‰

,“

e3, eA‰

and“

e3, e4

‰

to

s, using (1.16), and obtain

“

e4, eA‰

s “ e4eApsq ´ eApe4psqq “ 0

“

e4, eA‰

s “ pηA` ζAqe4psq ´ χABe

Bpsq “ η

A` ζA

“

e3, eA‰

s “ e3eApsq ´ eApe3psqq “ ´∇ AΩ

“

e3, eA‰

s “ pηA ´ ζAqe3psq ` ξAe4psq ´ χABeBpsq “ pηA ´ ζAqΩ` ξA,

“

e3, e4

‰

s “ e3e4psq ´ e4e3psq “ ´∇ 4Ω

“

e3, e4

‰

s “ 2ωe4psq ` 2pηB ´ ηBqeBpsq “ 2ω

40

Since e3pθAq “ bA, e4pθAq “ 0, eApθBq “ δAB we can apply“

e3, e4

‰

to θA, and obtain

“

e3, e4

‰

θA “ e3e4pθAq ´ e4e3pθAq “ ´DbA

“

e3, e4

‰

θA “ 2ωe4pθAq ` 2pηB ´ ηBqeBpθAq “ 2pηA ` ζAq

Using (1.2), we obtain the desired relation.

We now derive the equation for the metric g. Using (1.2), we obtain

DgAB “ ∇ 3gAB ` χAC gC

B` χ

BC gCA“ 2χ

AB“ 2pχ

AB` κgAB,

DgAB “ 2pχAB ` κgAB

In view of the formula for the projected Lie-derivative and the null frame (2.2),

DgAB “ 2ς´1BupgABq ` ΩBspgABq ` p∇ AbB `∇ BbAq

DgAB “ BspgABq

Combining the above, we obtain the desired relations.

In considering solutions pM,gq to the Einstein-Maxwell equations of the form

(2.1), we will add the above relations to the set of equations to linearize. In particular,

we use relations (2.4) to set ξ and ω to vanish and to substitute η in terms of ζ in

the equations. We will instead add equations (2.5)-(2.11) to the set of linearized

equations (see Section 4.2.2).

41

2.3 Transport equations for average quantities

Recall the definition of S-average given in (1.9). We specialize here to the foliation in

surfaces given by the Bondi gauge, and we derive the transport equations for average

quantities. They shall be used in Chapter 4 to derive the linearized equations for the

scalars involved.

To simplify the notation, we denote in the following S “ Su,s and r “ rpu, sq.

Proposition 2.3.0.1 (Proposition 2.2.9 in [34]). For any scalar function f , we have

∇ 4

ˆż

S

f

˙

“

ż

S

p∇ 4f ` κfq,

∇ 3

ˆż

S

f

˙

“

ż

S

p∇ 3f ` κfq ` Errr∇ 3p

ż

S

fqs

where the error term is given by the formula

Errr∇ 3p

ż

S

fqs : “ ´ς´1ς

ż

S

p∇ 3f ` κfq ` ς´1

ż

S

ςp∇ 3f ` κfq

`pΩ` ς´1Ωςq

ż

S

p∇ 4f ` κfq

´ς´1Ω

ż

S

ςp∇ 4f ` κfq ´ ς´1

ż

S

Ωςp∇ 4f ` κfq

In particular, we have

∇ 4r “r

2κ, ∇ 3r “

r

2pκ` Aq

where

A :“ ´ς´1κς ` κpΩ` ς´1Ωςq ` ς´1ς κ´ ς´1Ως κ´ ς´1Ωςκ

42

Proof. Recalling that e4 “ Bs, we compute

Bs

ˆż

S

f

˙

“

ż

S

pBsf ` gpDABs, eAqfq

We have DABs “ DAe4 and using the relations (1.15), we obtain

gpDABs, eAq “ gp´ζAe4 ` χACe

C , eAq “ κ

We easily deduce the desired relation along e4. The formula for derivative along e3 is

obtained in a similar way. See [34].

The equality for r follows by applying the Lemma to f “ 1.

Corollary 2.3.1 (Corollary 2.2.11 in [34]). For any scalar function f we have

∇ 4pfq “ ∇ 4f ` κf ,

∇ 4pfq “ ∇ 4f ´ κf

and

∇ 3pfq “ ∇ 3f ` Errr∇ 3pfqs

∇ 3pfq “ ∇ 3f ´ Errr∇ 3pfqs

where

Errr∇ 3pfqs : “ ´ς´1ςp∇ 3f ` κf ´ κfq ` ς´1pςp∇ 3f ` κfq ´ ς κfq

`pΩ` ς´1Ωςqp∇ 4f ` κf ´ κfq ´ ς´1Ωpςp∇ 4f ` κfq ´ ς κfq

´ς´1pΩςp∇ 4f ` κfq ´ Ωςκfq ` κf

43

Chapter 3

Reissner-Nordstrom spacetime

In this chapter, we introduce the Reissner-Nordstrom exterior metric, as well as rel-

evant background structure. For completeness, we collect here standard coordinate

transformations relevant to the study of Reissner-Nordstrom spacetime (see for ex-

ample [29]), even if not directly used in our proof.

We first fix in Section 3.1 an ambient manifold-with-boundaryM on which we de-

fine the Reissner-Nordstrom exterior metric gM,Q with parameters M and Q verifying

|Q| ăM . We shall then pass to more convenient sets of coordinates, like double null

coordinates, outgoing and ingoing Eddington-Finkelstein coordinates, and we shall

show how these sets of coordinates relate to the standard form of the metric as given

in (4).

In Section 3.2, we show that the Reissner-Nordstrom metric admits a Bondi form

as described in the previous chapter. We then describe the null frames associated to

such coordinates and the values of Ricci coefficients, curvature and electromagnetic

components.

Finally, in Section 3.3 we recall the symmetries of Reissner-Nordstrom spacetime

and present the main operators and commutation formulae. We also recall the main

44

properties of decomposition in spherical harmonics in Reissner-Nordstrom spacetime.

We will follow closely Section 4 of [16], where the main features of the Schwarz-

schild metric and differential structure are easily extended to the Reissner-Nordstrom

solution.

3.1 Differential structure and metric

We define in this section the underlying differential structure and metric in terms of

the Kruskal coordinates.

3.1.1 Kruskal coordinate system

Define the manifold with boundary

M :“ D ˆ S2 :“ p´8, 0s ˆ p0,8q ˆ S2 (3.1)

with coordinates pU, V, θ1, θ2q. We will refer to these coordinates as Kruskal coordi-

nates. The boundary

H` :“ t0u ˆ p0,8q ˆ S2

will be referred to as the horizon. We denote by S2U,V the 2-sphere tU, V u ˆ S2 ĂM

in M.

3.1.2 The Reissner-Nordstrom metric

We define the Reissner-Nordstrom metric on M as follows.

Fix two parameters M ą 0 and Q, verifying |Q| ăM . Let the function r :MÑ”

M `a

M2 ´Q2,8¯

be given implicitly as a function of the coordinates U and V

45

by

´UV “4r4`

pr` ´ r´q2

ˇ

ˇ

ˇ

r ´ r`r`

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

r´r ´ r´

ˇ

ˇ

ˇ

´

r´r`

¯2

exp´r` ´ r´

r2`

r¯

, (3.2)

where

r˘ “M ˘a

M2 ´Q2 (3.3)

We will also denote

rH “ r` “M `a

M2 ´Q2 (3.4)

Define also

ΥK pU, V q “r´r`

4rpU, V q2

´rpU, V q ´ r´r´

¯1`´

r´r`

¯2

exp´

´r` ´ r´r2`

rpU, V q¯

γAB “ standard metric on S2 .

Then the Reissner-Nordstrom metric gM,Q with parameters M and Q is defined to

be the metric:

gM,Q “ ´4ΥK pU, V q dUdV ` r2pU, V q γABdθ

AdθB. (3.5)

Note that the horizon H` “ BM is a null hypersurface with respect to gM,Q. We

will use the standard spherical coordinates pθ1, θ2q “ pθ, φq, in which case the metric

γ takes the explicit form

γ “ dθ2` sin2 θdφ2. (3.6)

The above metric (3.5) can be extended to define the maximally-extended Reissner-

46

Nordstrom solution on the ambient manifold p´8,8q ˆ p8,8q ˆ S2. In this paper,

we will only consider the manifold-with-boundary M, corresponding to the exterior

of the spacetime.

The Reissner-Nordstrom family of spacetimes pM,gM,Qq is the unique electrovac-

uum spherically symmetric spacetime. It is a static and asymptotically flat spacetime.

The parameter Q may be interpreted as the charge of the source. This metric clearly

reduces to Schwarzschild spacetime when Q “ 0, therefore M can be interpreted as

the mass of the source.

Using definition (3.5), the metric gM,Q is manifestly smooth in the whole domain.

We will now describe different sets of coordinates for which smoothness breaks down,

but which are nevertheless useful for computations.

3.1.3 Double null coordinates u, v

We define another double null coordinate system that covers the interior ofM, modulo

the degeneration of the angular coordinates. This coordinate system, pu, v, θ1, θ2q, is

called double null coordinates and are defined via the relations

U “ ´2r2`

r` ´ r´exp

ˆ

´r` ´ r´

4r2`

u

˙

and V “2r2`

r` ´ r´exp

ˆ

r` ´ r´4r2`

v

˙

. (3.7)

Using (3.7), we obtain the Reissner-Nordstrom metric on the interior ofM in pu, v, θ1, θ2q-

coordinates:

gM,Q “ ´4Υ pu, vq du dv ` r2pu, vq γABdθ

AdθB (3.8)

47

with

Υ :“ 1´2M

r`Q2

r2(3.9)

and the function r : p´8,8q ˆ p´8,8q Ñ´

M `a

M2 ´Q2,8¯

defined implicitly

via the relations between pU, V q and pu, vq. In pu, v, θ1, θ2q-coordinates, the horizon

H` can still be formally parametrised by p8, v, θ1, θ2q with v P R, pθ1, θ2q P S2.

Note that u, v are regular optical functions. Their corresponding null geodesic

generators are

L :“ ´gabBavBb “1

ΥBu, L :“ ´gabBauBb “

1

ΥBv, (3.10)

They verify

gpL,Lq “ gpL,Lq “ 0, gpL,Lq “ ´2Υ´1, DLL “ DLL “ 0.

3.1.4 Standard coordinates t, r

Recall the form of the metric (3.8) in double null coordinates. Setting

t “ u` v

we may rewrite the above metric in coordinates pt, r, θ, φq in the usual form (4):

gM,Q “ ´Υprqdt2 `Υprq´1dr2` r2

pdθ2` sin2 θdφ2

q, (3.11)

48

which covers the interior of M. Observe that

Υprq :“ 1´2M

r`Q2

r2“pr ´ r´qpr ´ r`q

r2

where r´ and r` are defined in (3.3).

The null vectors L and L defined in (3.10), in pt, rq coordinates can be written as

L “ Υ´1Bt ´ Br, L “ Υ´1

Bt ` Br, (3.12)

3.1.5 Ingoing Eddington-Finkelstein coordinates v, r

We define another coordinate system that covers the interior of M. This coordinate

system, pv, r, θ, φq is called ingoing Eddington-Finkelstein coordinates and makes use

of the above defined functions v and r. The Reissner-Nordstrom metric on the interior

of M in pv, r, θ, φq-coordinates is given by

gM,Q “ ´Υprqdv2` 2dvdr ` r2

pdθ2` sin2 θdφ2

q. (3.13)

3.2 The Bondi form of the Reissner-Nordstrom

metric

We define here another coordinate system that covers the topological interior of the

manifoldM, and which achieves the Bondi form of the Reissner-Nordstrom metric as

described in Chapter 2. These coordinate system covers therefore the open exterior

of the Reissner-Nordstrom black hole spacetime.

Recall the function r implicitly defined by (3.2) and the function u defined by

(3.7).

49

In the coordinate system pu, r, θ, φq, called outgoing Eddington-Finkelstein coor-

dinates, the Reissner-Nordstrom metric on the interior of M is given by

ds2“ ´2dudr ´Υprqdu2

` r2pdθ2

` sin2 θdφ2q. (3.14)

Notice that this metric is of the Bondi form 2.1 with the coordinate function s “ r1

and

ς “ 1, Ω “ ´Υ, bA “ 0, gAB “ r2γAB (3.15)

The normalized outgoing geodesic null frame N associated to the above is given

by

e3 “ 2Bu ` ΩBr, e4 “ Br (3.16)

together with a local frame field pe1, e2q on Su,r.

The above frame does not extend regularly to the horizon H`, while the rescaled

null frame

N˚ “ tΩ´1e3, Ωe4u

extends regularly to a non-vanishing null frame on H`.

We will always compute with respect to the normalized null frame N , but nev-

ertheless passing to N˚ will be useful to understand which quantities are regular on

the horizon.

1Notice that rpu, sq “ s verifies the definition given by (2.3), since at u “ constant and s “constant, the metric g induced on Su,s is given by r2

`

dθ2 ` sin2 θdφ2˘

which verifies |Su,s| “ 4πr2.

50

3.2.1 Ricci coefficients and curvature components

We recall here the connection coefficients, curvature and electromagnetic components

with respect to the null frame (3.16).

The Ricci coefficients are given by

pχAB “ pχAB“ 0, η “ η “ ξ “ ξ “ ζ “ 0 ω “ 0 (3.17)

κ “2

r, κ “

2Ω

r“ ´

2

r

ˆ

1´2M

r`Q2

r2

˙

, ω “M

r2´Q2

r3(3.18)

Remark 3.2.1. As opposed to the Ricci coefficients in double null gauge used in [16],

in the Bondi gauge all the quantities are regular near the horizon H`.

The electromagnetic components are given by

pF qβ “ pF qβ “ 0, pF qσ “ 0, pF qρ “Q

r2(3.19)

The curvature components are given by

α “ α “ 0, β “ β “ 0, σ “ 0, ρ “ ´2M

r3`

2Q2

r4(3.20)

We also have that

K “1

r2(3.21)

for the Gauss curvature of the round S2-spheres.

Recalling the definition for a scalar function (1.10), the above values in particular

51

imply for the scalar functions which do not vanish in Reissner-Nordstrom:

κ “ κ “ ω “ ˇpF qρ “ ρ “ K “ 0 (3.22)

H` I `

Στ

Σ0

Mp0, τq

Figure 3.1: Foliation Στ in the Penrose diagram of Reissner-Nordstrom spacetime

3.3 Reissner-Nordstrom symmetries and operators

In this section, we recall the symmetries of the Reissner-Nordstrom metric, and spe-

cialize the operators discussed in Section 1.1.2 to the Reissner-Nordstrom metric in

the Bondi form (3.14).

3.3.1 Killing fields of the Reissner-Nordstrom metric

We discuss the Killing fields associated to the metric gM,Q. Notice that the Reissner-

Nordstrom metric possesses the same symmetries as the ones possessed by Schwarz-

schild spacetime.

We define the vectorfield T to be the timelike Killing vector field Bt of the pt, rq

coordinates in (3.11). In outgoing Eddington-Finkelstein coordinates is given by

T “ Bu

52

The vector field extends to a smooth Killing field on the horizon H`, which is

moreover null and tangential to the null generator of H`.

In terms of the null frames defined above, the Killing vector field T can be written

as

T “1

2pΥe˚3 ` e

˚4q “

1

2pe3 `Υe4q (3.23)

Notice that at on the horizon, T corresponds up to a factor with the null vector of

N ˚ frame, T “ 12e˚4 .

We can also define a basis of angular momentum operator Ωi, i “ 1, 2, 3. Fixing

standard spherical coordinates on S2, we have

Ω1 “ Bφ, Ω2 “ ´ sinφBθ ´ cot θ cosφBφ, Ω3 “ cosφBθ ´ cot θ sinφBφ

The Lie algebra of Killing vector fields of gM,Q is then generated by T and Ωi, for

i “ 1, 2, 3.

3.3.2 The Su,r-tensor algebra in Reissner-Nordstrom

We now specialize the general definitions of the projected Lie and covariant differential

operators of Section 1.1.2 to the Reissner-Nordstrom metric with null directions given

by (3.16).

If ξ is a Su,r tensor of rank n on pM,gM,Qq we have in components

pDξqA1,...An “ BrpξA1,...Anq, pDξqA1,...An “ 2BupξA1,...Anq ` ΩBrpξA1,...Anq (3.24)

Since χ, χ only have a trace-component in Reissner-Nordstrom, one can specialize

53

formulas (1.2) as

p∇ 4ξqA “ BrpξAq ´1

2κξA, p∇ 3ξqA “ 2BupξAq ` ΩBrpξAq ´

1

2κξA (3.25)

p∇ 4ξqA“ Brpξ

Aq `

1

2κξA, p∇ 3ξq

A“ 2Bupξ

Aq ` ΩBrpξ

Aq `

1

2κξA (3.26)

for 1-forms and 1-vectors and

p∇ 4θqAB “ BrpθABq ´ κθAB, p∇ 3θqAB “ 2BupθABq ` ΩBrpθABq ´ κθAB (3.27)

p∇ 4θqAB“ Brpθ

ABq ` κθAB, p∇ 3θq

AB“ 2BupθABq ` ΩBrpθABq ` κθAB (3.28)

for symmetric traceless 2-tensors.

3.3.3 Commutation formulae in Reissner-Nordstrom

Adapting the commutation formulae (1.16) to the Reissner-Nordstrom metric, we

obtain the following commutation formulae. For projected covariant derivatives for

ξ “ ξA1...An any n-covariant S2u,r-tensor in Reissner-Nordstrom metric pM,gM,Qq in

Bondi gauge we have

∇ 3 ∇BξA1...An ´ ∇B ∇3ξA1...An “ ´1

2κ ∇BξA1...An ,

∇4 ∇BξA1...An ´ ∇B ∇4ξA1...An “ ´1

2κ ∇BξA1...An , (3.29)

∇3 ∇4ξA1...An ´ ∇4 ∇3ξA1...An “ 2ω ∇4ξA1...An .

In particular, we have

“

∇4, r ∇A

‰

ξ “ 0 ,“

∇3, r ∇A

‰

ξ “ 0 . (3.30)

54

We summarize here the commutation formulae for the angular operators defined in

Section 1.1.2. Let ρ, σ be scalar functions, ξ be a 1-tensor and θ be a symmetric

traceless 2-tensor in Reissner-Nordstrom manifold. Then:

r∇ 4, D1s ξ “ ´1

2κD1ξ, r∇ 3, D1s ξ “ ´

1

2κD1ξ (3.31)

r∇ 4, D‹1s pρ, σq “ ´1

2κD‹1pρ, σq, r∇ 3, D‹1s pρ, σq “ ´

1

2κD‹1pρ, σq, (3.32)

r∇ 4, D2s θ “ ´1

2κD2θ, r∇ 3, D2s θ “ ´

1

2κD2θ (3.33)

r∇ 4, D‹2s ξ “ ´1

2κD‹2ξ, r∇ 3, D‹2s ξ “ ´

1

2κD‹2ξ (3.34)

3.3.4 The l “ 0, 1 spherical harmonics

We collect some known definitions and properties of the Hodge decomposition of

scalars, one forms and symmetric traceless two tensors in spherical harmonics. We

also recall some known elliptic estimates. See Section 4.4 of [16] for more details.

The l “ 0, 1 spherical harmonics and tensors supported on l ě 2

We denote by 9Y lm, with |m| ď l, the well-known spherical harmonics on the unit

sphere, i.e.

4 09Y lm “ ´lpl ` 1q 9Y l

m

where4 0 denotes the laplacian on the unit sphere S2. The l “ 0, 1 spherical harmonics

are given explicitly by

9Y l“0m“0 “

1?

4π, (3.35)

9Y l“1m“0 “

c

3

8πcos θ, 9Y l“1

m“´1 “

c

3

4πsin θ cosφ, 9Y l“1

m“1 “

c

3

4πsin θ sinφ(3.36)

55

This family is orthogonal with respect to the standard inner product on the sphere,

and any arbitrary function f P L2pS2q can be expanded uniquely with respect to such

a basis.

In the foliation of Reissner-Nordstrom spacetime, we are interested in using the

spherical harmonics with respect to the sphere of radius r. For this reason, we nor-

malize the definition of the spherical harmonics on the unit sphere above to the

following.

We denote by Y lm, with |m| ď l, the spherical harmonics on the sphere of radius

r, i.e.

4Y lm “ ´

1

r2lpl ` 1qY l

m

where4 denotes the laplacian on the sphere Su,r of radius r. Such spherical harmonics

are normalized to have L2 norm in Su,r equal to 1, so they will in particular be given by

Y lm “

1r

9Y lm. We use these basis to project functions on Reissner-Nordstrom manifold

in the following way.

Definition 3.3.1. We say that a function f on M is supported on l ě 2 if the

projections

ż

Su,r

f ¨ Y lm “ 0

vanish for Y l“1m for m “ ´1, 0, 1. Any function f can be uniquely decomposed orthog-

onally as

f “ cpu, rqY l“0m“0 `

1ÿ

i“´1

cipu, rqYl“1m“ipθ, ϕq ` flě2 (3.37)

where flě2 is supported in l ě 2.

56

In particular, we can write the orthogonal decomposition

f “ fl“0 ` fl“1 ` flě2

where

fl“0 “1

4πr2

ż

Su,r

f (3.38)

fl“1 “

1ÿ

i“´1

˜

ż

Su,r

f ¨ Y l“1m“i

¸

Y l“1m“i (3.39)

Recall that an arbitrary one-form ξ on Su,r has a unique representation ξ “

rD‹1pf, gq, for two uniquely defined functions f and g on the unit sphere, both with

vanishing mean, i.e. fl“0 “ gl“0 “ 0. In particular, the scalars div ξ and curl ξ are

supported in l ě 1. As in [16], we define

Definition 3.3.2. We say that a smooth Su,r one form ξ is supported on l ě 2 if the

functions f and g in the unique representation

ξ “ rD‹1pf, gq

are supported on l ě 2. Any smooth one form ξ can be uniquely decomposed orthogo-

nally as

ξ “ ξl“1 ` ξlě2

where the two scalar functions rD1ξ “ prdiv ξl“1, rcurl ξl“1q are in the span of (3.35)

and ξlě2 is supported on l ě 2.

Recall that an arbitrary symmetric traceless two-tensors θ on Su,r has a unique

57

representation

θ “ r2D‹2D‹1pf, gq

for two uniquely defined functions f and g on the unit sphere, both supported in

l ě 2. In particular, the scalars div div θ and curl div θ are supported in l ě 2.

For future reference, we recall the following lemma.

Lemma 3.3.4.1 (Lemma 4.4.1 in [16]). The kernel of the operator T “ r2D‹2D‹1 is

finite dimensional. More precisely, if the pair of functions pf1, f2q is in the kernel,

then

f1 “ cY l“0m“0 `

1ÿ

i“´1

ciYl“1m“ipθ, ϕq, f2 “ cY l“0

m“0 `

1ÿ

i“´1

ciYl“1m“ipθ, ϕq

for constants c, ci, c, ci.

Elliptic estimates

Consider a one-form ξ on M and its decomposition ξ “ ξl“1 ` ξlě2 as in Definition

3.3.2. Then Proposition 1.1.2.1 implies following elliptic estimate.

Lemma 3.3.4.2. Let ξ be a one-form on M. Then there exists a constant C ą 0

such that the following estimate holds:

ż

S

|ξ|2 ď C

ˆż

S

|rdiv ξl“1|2` |rcurl ξl“1|

2` |rD‹2ξ|2

˙

Proof. Using the orthogonal decomposition of ξ, we have

ż

S

|ξ|2 “

ż

S

|ξl“1|2`

ż

S

|ξlě2|2

Observe that, according to Lemma 3.3.4.1, ξlě2 is in the kernel of D‹2. Applying (1.4)

58

to ξl“1 and (1.7) to ξlě2 we obtain the desired estimate.

Average and check quantities

Here we collect the useful properties associated to the decomposition in average and

check quantities.

Lemma 3.3.4.3. Any scalar function f :MÑ R verifies

f lě1 “ 0, fl“0 “ 0 (3.40)

Therefore f “ f l“0 ` flě1.

Proof. Using (1.9) and (3.37), we compute

|S|f “

ż

S

˜

cpu, rqY l“0m“0 `

l“1ÿ

i“´1

cipu, rqYl“1m“ipθ, ϕq ` flě2

¸

sin θdθdϕ

“ |S|cpu, rqY l“0m“0 `

l“1ÿ

i“´1

cipu, rq

ż

S

`

Y l“1m“ipθ, ϕq ` flě2

˘

sin θdθdϕ

and recalling that, by orthogonality of the spherical harmonics,

ż

S

Y lmpθ, ϕqY

l1

m1pθ, ϕq sin θdθdϕ “ δll1δmm1

the integral on the right hand side vanishes. Therefore f lě1 “ 0 and fl“0 “ f l“0. On

the other hand,

f “ f ´ f “ cpu, sqY l“0m“0 `

l“1ÿ

i“´1

cipu, sqYl“1m“ipθ, ϕq ` flě2 ´ cpu, sqY

l“0m“0

“

l“1ÿ

i“´1

cipu, sqYl“1m“ipθ, ϕq ` flě2

59

therefore f is supported in l ě 1.

We derive the transport equation for the projection to the l “ 1 spherical har-

monics of a function f on M.

Lemma 3.3.4.4. Let f be a scalar function on M. Then

∇ 4pfl“1q “ p∇ 4fql“1

∇ 3pfl“1q “ p∇ 3fql“1

Proof. Applying ∇ 4 “ Br to the expression for the projection to the l “ 1 spherical

harmonics given by (3.38), we obtain

∇ 4pfl“1q “

1ÿ

i“´1

∇ 4

´

ˆż

S

f ¨ Y l“1m“i

˙

Y l“1m“i

¯

Recall that the normalized spherical harmonics are defined as Y l“1m “ 1

r9Y l“1m , where

9Y l“1m are given by (3.36), and therefore ∇ 4p 9Y l“1

m q “ 0. This implies

∇ 4pYl“1m q “ ∇ 4

ˆ

1

r9Y l“1m

˙

“ ∇ 4

ˆ

1

r

˙

9Y l“1m “ ´

1

2rκ 9Y l“1

m “ ´1

2κY l“1

m

where we used Proposition 2.3.0.1.

60

Using again Proposition 2.3.0.1, the computation gives

∇ 4pfl“1q “

1ÿ

i“´1

∇ 4

ˆż

S

f ¨ Y l“1m“i

˙

Y l“1m“i `

1ÿ

i“´1

ˆż

S

f ¨ Y l“1m“i

˙

∇ 4Yl“1m“i

“

1ÿ

i“´1

ˆż

S

∇ 4pf ¨ Yl“1m“iq ` κf ¨ Y

l“1m“i

˙

Y l“1m“i `

1ÿ

i“´1

ˆż

S

f ¨ Y l“1m“i

˙

p´1

2κY l“1

m q

“

1ÿ

i“´1

ˆż

S

∇ 4pfq ¨ Yl“1m“i ` f ¨∇ 4pY

l“1m“iq `

1

2κf ¨ Y l“1

m“i

˙

Y l“1m“i

“

1ÿ

i“´1

ˆż

S

∇ 4pfq ¨ Yl“1m“i

˙

Y l“1m“i “ p∇ 4fql“1

as desired. Similarly for ∇ 3f .

61

Chapter 4

The linearized gravitational and

electromagnetic perturbations

around Reissner-Nordstrom

In this chapter, we present the equations of linearized gravitational and electromag-

netic perturbations around Reissner-Nordstrom.

In Section 4.1 we describe the procedure to the linearization of the equations of

Section 1.3. In Section 4.2 we summarize the complete set of equations describing

the dynamical evolution of a linear perturbation of Reissner-Nordstrom spacetime.

4.1 A guide to the formal derivation

We give in this section a formal derivation of the system from the equations of Section

1.3 and of Section 2.2.

62

4.1.1 Preliminaries

We identify the general manifoldM and its Bondi coordinates pu, s, θ1, θ2q of Section

2.1 with the interior of the Reissner-Nordstrom spacetime in its Bondi form in Section

3.2.

OnM, we consider a one-parameter family of Lorentzian metrics gpεq of the form

(2.1). More precisely:

gpεq “ ´2ςpεqduds` ςpεq2Ωpεqdu2

` gABpεq

ˆ

dθA ´1

2ςpεqbpεqAdu

˙ˆ

dθB ´1

2ςpεqbpεqBdu

˙ (4.1)

such that gp0q “ gM,Q expressed in the outgoing Eddington-Finkelstein coordinates

(3.14), i.e.

ςp0q “ 1, Ωp0q “ ´

ˆ

1´2M

r`Q2

r2

˙

, bAp0q “ 0, gABp0q “ r2γAB

In view of the general discussion in Section 2.1, associated to the metric (4.1)

there is an associated family of normalized frames of the form

e3 “ 2ς´1pεqBu ` ΩpεqBs ` bpεq

ABθA , e4 “ Bs, eA “ BθA

Note that this frame does not extend smoothly to the event horizon H`. On the

other hand, the rescaled null frame

Ω´1pεqe3, Ωpεqe4

is smooth up to the horizon.

63

4.1.2 Outline of the linearization procedure

We now linearize the smooth one-parameter family of metrics (4.1) in terms of ε. We

linearize the full system of equations obtained in Section 1.3 around the values of the

connection coefficients and curvature components in Reissner-Nordstrom obtained in

Section 3.2.1. We describe the outline of the procedure in few different cases.

Linearization of one forms and two tensors

From (3.17), (3.19), (3.20), we notice that all the one-forms and symmetric traceless 2-

tensors appearing in the Einstein-Maxwell equations of Section 1.3 vanish in Reissner-

Nordstrom.

Formally, we have

pχpεq “ 0` pχ

pχpεq “ 0` pχ

ηpεq “ 0` η

ξpεq “ 0` ξ

ζpεq “ 0` ζ

pF qβpεq “ 0` pF qβ

pF qβpεq “ 0` pF qβ

αpεq “ 0` α

αpεq “ 0` α

βpεq “ 0` β

βpεq “ 0` β

64

The linearization of the equations involving the above tensors simply consists in

discarding terms containing product of those, while keeping the other terms. In

doing so, we will make sure to include the information obtained by the equation (2.4)

for the Bondi form of the metric.

To give an example, consider equations (1.22):

∇ 3pχ` κ pχ` 2ωpχ “ ´2D‹2ξ ´ α ` 2pη ` η ´ 2ζqpbξ

∇ 4pχ` κ pχ` 2ωpχ “ ´2D‹2ξ ´ α ` 2pη ` η ` 2ζqpbξ

In linearizing them, we observe that the term 2pη ` η ´ 2ζqpbξ is quadratic, and

ω “ ξ “ 0 by (2.4). We therefore obtain

∇ 3pχ` κ pχ` 2ωpχ “ ´2D‹2ξ ´ α

∇ 4pχ` κ pχ “ ´α

which appear on (4.17) and (4.18).

In this way we linearize (1.22), (1.24), (1.25), (1.27), (1.28), (1.29), (1.31), (1.36),

(1.38), (1.39).

Linearization of scalar functions

Recall the non-vanishing values of the scalars κ, κ, ω, pF qρ, ρ and K in Reissner-

Nordstrom given by (3.17), (3.19), (3.20), (3.21). Nevertheless, by (3.22) all check-

quantities vanish. Moreover, the scalars σ and pF qσ vanish.

We take advantage of this fact by using the decomposition into average and check

as defined in (1.10).

65

For instance, we write

κpεq “2

r`

ˆ

κpεq ´2

r

˙

` κpεq

“2

r`

(1)

κpεq ` κpεq

where we define(1)

κpεq “ κpεq ´ 2r.

We therefore define two scalar functions for each non-vanishing scalar: the average

to which we subtract the value in Reissner-Nordstrom (denoted by a superscript p1q)

and the check quantity.

In particular, we define

(1)

κpεq “ κpεq ´2

r,

(1)

κpεq “ κpεq `2

r

ˆ

1´2M

r`Q2

r2

˙

,

(1)

ωpεq “ ωpεq ´

ˆ

M

r2´Q2

r3

˙

,

(1)

pF qρpεq “ pF qρpεq ´Q

r2

(1)

pF qσpεq “ pF qσpεq

(1)

ρpεq “ ρpεq `2M

r3´

2Q2

r4

σpεq “ σpεq

(1)

Kpεq “ Kpεq ´1

r2

The check quantities linearize in the obvious way.

In linearizing the equations for κ, we will obtain equations for the quantities(1)

κ

and κ. To simplify the notation, we can therefore denote κ the value of the quantity

in Reissner-Nordstrom. This gives(1)

κ “ κ´ κ. Similarly for all the other quantities.

66

In Section 2.3, we computed the transport equations of average quantities. Using

those, we compute the equations for the linearized quantities above.

For instance, consider equation (1.23):

∇ 4κpεq `1

2κpεq2 “ ´pχpεq ¨ pχpεq ´ 2 pF qβpεq ¨ pF qβpεq

The right hand side is quadratic, therefore in linearizing we have

∇ 4κpεq `1

2κpεq2 “ 0

We can use Corollary 2.3.1, to compute ∇ 4pκpεqq:

∇ 4pκpεqq “ ∇ 4κpεq “ ´1

2κpεq2 “ ´

1

2κpεq

2“ ´

1

2p

(1)

κpεq ` κq2 “ ´κ(1)

κpεq ´1

2κ2

On the other hand, using Proposition 2.3.0.1, we have

∇ 4

ˆ

2

r

˙

“ ´2

r2∇ 4r “ ´

1

rκpεq

Therefore, writing κ “ κpεq ´(1)

κpεq, we have

∇ 4p(1)

κpεqq “ ∇ 4pκpεqq ´∇ 4

ˆ

2

r

˙

“ ´κ(1)

κpεq ´1

2κ2`

1

rκpεq

“ ´κ(1)

κpεq ´1

2κpκpεq ´

(1)

κpεqq `1

rκpεq

“ ´1

2κ

(1)

κpεq `

ˆ

´1

2κ`

1

r

˙

κpεq “ ´1

2κ

(1)

κpεq

which gives equation (4.28). All the other equations are obtained in a similar manner.

The equation for the check part for a scalar quantity is obtained applying again

67

Corollary 2.3.1.

It seems that we have doubled the equations involving scalar quantities. In reality,

the separation between(1)

f and f for a scalar quantity f reflects the projection into

spherical harmonics. Indeed, we have that(1)

fpεqlě1 “ 0 and fpεql“0 “ 0, where the

projections are intended to be with respect to the Reissner-Nordstrom metric. This

is proved in the following way. Since fpεq has vanishing mean with respect to the

metric gpεq, we have D‹1pεqpfpεq, 0q “ 0 and therefore

0 “ D‹1pεqpfpεq, 0q “ pD‹1p0q ` εqp(1)

fpεq ` fM,Q, 0q “ D‹1p0qp(1)

fpεq, 0q `Opε2q

where D‹1p0q is the angular operator D‹1 in Reissner-Nordstrom. Similarly, in taking

the mean of fpεq we see that it has vanishing mean with respect to the Reissner-

Nordstrom spacetime, modulo quadratic terms.

In this way, we linearize (1.23), (1.26), (1.30), (1.32), (1.34), (1.37), (1.41), (1.42).

Linearization of metric coefficients

We now outline the linearization of the metric coefficients in Bondi form, verifying

the equations given by Lemma 2.2.0.1. The metric coefficients are ςpεq, Ωpεq, bpεq and

gpεq.

We decompose the scalar functions ςpεq and Ωpεq as above. We define

(1)

Ωpεq “ Ωpεq `

ˆ

1´2M

r`Q2

r2

˙

and define ςpεq “ ςpεq ´ ςpεq, Ωpεq “ Ωpεq ´ Ωpεq. Since e3puq “ 2ς´1 and u is equal

to its average,(1)

ςpεq “(1)

ςpεq ´ 1 “ 0. Since e3psq “ Ω, we obtain(1)

Ω “ e3ps ´ rq ` 12r

(1)

κ.

68

As a scalar in Reissner-Nordstrom metric where s “ r we have

(1)

κ “ κ(1)

Ω (4.2)

The vector b vanishes on Reissner-Nordstrom, therefore the linearization of (2.9)

is straightforward.

We now show how to linearize the equations for the metric gpεq (2.10) and (2.11).

Since gp0q “ r2γAB, we decompose g into:

gAB “1

2ptrγgqγAB ` gAB (4.3)

where the trace and the traceless part are computed in terms of the round sphere

metric, i.e.

trγg “ γABgAB, γAB gAB “ 0

Plugging in the decomposition (4.3) in the equations for the metric (2.10), we obtain

Bsp1

2ptrγgqγAB ` gABq “ 2pχAB ` κ

ˆ

1

2ptrγgqγAB ` gAB

˙

(4.4)

Recalling that BspγABq “ 0 in Reissner-Nordstrom background, the left hand side of

(4.4) becomes

Bsp1

2ptrγgqγAB ` gABq “

1

2BsptrγgqγAB ` Bs gAB

69

Observe that Bs gAB is traceless with respect to γ, since

0 “ BsppγqABgABq “ pγq

ABBs gAB

The right hand side of (4.4) is given by

1

2κptrγgqγAB ` 2pχAB ` κgAB

Observe that pχ is traceless with respect to γ modulo quadratic terms, therefore sep-

arating the equation into its traceless and trace part we obtain:

Bs gAB ´ κgAB “ 2pχAB,

Bsptrγgq “ κptrγgq

Using (3.27), we have

∇ 4 gAB “ 2pχAB

∇ 4ptrγgq “ κptrγgq

Define

(1)

trg “ trγg ´ 2r2

trγg “ trγg ´ trγg

70

By Corollary 2.3.1, we have

∇ 4p(1)

trgq “ ∇ 4ptrγgq ´ 2∇ 4pr2q “ ∇ 4ptrγgq ´ 2r2κ “ κtrγg ´ 2r2κ “ κ

(1)

trg,

∇ 4ptrγgq “ ∇ 4ptrγgq ´∇ 4ptrγgq “ pκ´ div bqptrγgq ´ κtrγg “ κtrγg ` 2r2κ´ 2r2div b

Similarly, for ∇ 3 we have

∇ 3 gAB “ 2pχAB` 2pD‹2bqAB,

∇ 3p(1)

trgq “ κ(1)

trg,

∇ 3ptrγgq “ 2r2κ´ 2r2div b´ 2r2κΩ

Using that r2∇ 3pr´2fq “ ∇ 3f ´ κf , we obtain the equations for the metric compo-

nents.

Linearization of Gauss curvature

We linearize the Gauss curvatureKpεq of the metric g as for the above scalar functions.

We define

(1)

Kpεq “ Kpεq ´1

r2

Kpεq “ Kpεq ´Kpεq

Observe that, by Gauss-Bonnet theorem,ş

SKpεq “ 4π, therefore it implies that

Kpεq “ 14πr2

ş

SKpεq “ 1

r2, and consequently

(1)

Kpεq “ 0.

In general, the linearization of the Gauss curvature for a metric is given by

2pδKq “ ´1

24 ptrpδgqq ´Ktrpδgq ` div div ypδgq

71

Writing (4.3) as

gAB “ r2γAB ` δg “ r2γAB `1

2ptrγg ´ 2r2

qγAB ` gAB

we obtain

2

ˆ

Kpεq ´1

r2

˙

“ ´1

24`

trγgpεq ´ 2r2˘

´1

r2

`

trγgpεq ´ 2r2˘

` div div gpεq

Projecting into the l “ 0 mode, since(1)

K “ 0, this implies(1)

trg “ 0. The projection to

the l ě 1 mode gives

2K “ ´1

24´

trγg¯

´1

r2

´

trγg¯

` div div g (4.5)

In particular, projecting to the l “ 1 mode, we obtain that`

124 ` 1

r2

˘

trγgl“1“ 0 and

div div gl“1“ 0, therefore

Kl“1 “ 0 (4.6)

The vanishing of the l “ 1 spherical harmonics of the Gauss curvature will be crucial

later in the proof of linear stability for the lower mode of the perturbations.

4.2 The full set of linearized equations

In the following, we present the equations arising from the formal linearisation out-

lined above.

72

4.2.1 The complete list of unknowns

The equations will concern the following set of quantities, separated into symmet-

ric traceless 2-tensors, one-tensors and scalar functions on the Reissner-Nordstrom

manifold pM,gM,Qq.

S2 “ tαAB, αAB, pχAB, pχAB, gABu

S1 “ tζA, ηA, ξA, βA, β

A, pF qβA,

pF qβA, bAu

S0 “ tκ, κ, ω, ρ, σ, pF qρ, pF qσ, trγg, Ω, ς , K

(1)

κ,(1)

κ,(1)

ω,(1)

ρ,(1)

pF qρ,(1)

pF qσ,(1)

Ωu

Definition 4.2.1. We say that S “ S0YS1YS2 is a linear gravitational and

electromagnetic perturbation around Reissner-Nordstrom spacetime if the

quantities in S satisfy the equations (4.7)-(4.63) below.

Observe that in the definition we omitted(1)

σ (indeed,(1)

σ “ 0 is the linearization of

(1.27)),(1)

K and(1)

trg, as they are implied to be zero by the previous subsection.

In what follows, the scalar functions without any superscript or check are to be

intended as quantities in the background spacetime pM,gM,Qq.

4.2.2 Equations for the linearised metric components

The linearization of (2.10) and (2.11) for 2-tensors are the following:

∇ 4 g “ 2pχ, (4.7)

∇ 3 g “ 2pχ` 2D‹2b (4.8)

73

The linearization of (2.5), (2.7) and (2.9) are the following:

D‹1pς , 0q “ ζ ´ η (4.9)

D‹1pΩ, 0q “ ξ ` Ωpη ´ ζq (4.10)

∇ 4b´1

2κb “ ´2pη ` ζq (4.11)

The linearization of (2.6), (2.8), (2.10) and (2.11) are the following:

(1)

κ “ κ(1)

Ω (4.12)

and

∇ 4ς “ 0 (4.13)

∇ 4Ω “ ´2ω, (4.14)

∇ 4

´

r´2trγg

¯

“ 2κ (4.15)

∇ 3

´

r´2trγg

¯

“ 2pκ´ κΩq ´ 2div b (4.16)

4.2.3 Linearized null structure equations

We collect here the linearisation of the equations in Section 1.3.2.

The linearization of (1.22) and (1.25) are the following:

∇ 3pχ` pκ` 2ωq pχ “ ´2D‹2ξ ´ α, (4.17)

∇ 4pχ` κ pχ “ ´α, (4.18)

∇ 3pχ`

ˆ

1

2κ´ 2ω

˙

pχ “ ´2 D‹2η ´1

2κpχ (4.19)

∇ 4pχ`1

2κ pχ “ 2 D‹2ζ ´

1

2κpχ, (4.20)

74

The linearisation of (1.28), (1.29) and (1.31) are the following:

∇ 3ζ `

ˆ

1

2κ´ 2ω

˙

ζ “ 2D‹1pω, 0q ´ˆ

1

2κ` 2ω

˙

η `1

2κξ ´ β ´ pF qρ pF qβ,(4.21)

∇ 4ζ ` κζ “ ´β ´ pF qρ pF qβ, (4.22)

∇ 4ξ `1

2κξ “ ´2D‹1pω, 0q ` 2ωpη ´ ζq, (4.23)

∇ 4η `1

2κη “ ´

1

2κζ ´ β ´ pF qρ pF qβ, (4.24)

div pχ “ ´1

2κζ ´

1

2D‹1pκ, 0q ` β ´ pF qρ pF qβ, (4.25)

div pχ “1

2κζ ´

1

2D‹1pκ, 0q ´ β ` pF qρ pF qβ (4.26)


∇ 3(1)

κ`1

2κ

(1)

κ “ 2ω(1)

κ`4

r

(1)

ω ` 2(1)

ρ, (4.27)

∇ 4(1)

κ`1

2κ

(1)

κ “ 0, (4.28)

∇ 3(1)

κ`1

2κ

(1)

κ “ ´2κ(1)

ω, (4.29)

∇ 4(1)

κ`1

2κ

(1)

κ “

ˆ

2M

r2´

2Q2

r3

˙

(1)

κ` 2(1)

ρ, (4.30)

∇ 4(1)

ω “(1)

ρ`2Q

r2

(1)

pF qρ`

ˆ

M

r2´

3Q2

2r3

˙

(1)

κ (4.31)

and

∇ 4κ` κκ “ 0, (4.32)

∇ 3κ`

ˆ

1

2κ´ 2ω

˙

κ “ ´1

2κ`

κ´ κΩ˘

` 2κω ` 2div η ` 2ρ, (4.33)

∇ 4κ`1

2κκ “ ´

1

2κκ´ 2div ζ ` 2ρ, (4.34)

∇ 3κ` pκ` 2ωq κ “ ´2κω ` 2div ξ `

ˆ

1

2κκ´ 2ρ

˙

Ω (4.35)

∇ 4ω “ ρ` 2 pF qρ ˇpF qρ, (4.36)

75


0 “ ´1

4κ

(1)

κ´1

4κ

(1)

κ´(1)

ρ` 2 pF qρ(1)

pF qρ (4.37)

and

curl ξ “ 0 (4.38)

σ “ curl ζ (4.39)

curl pζ ´ ηq “ 0 (4.40)

K “ ´1

4κκ´

1

4κκ´ ρ` 2 pF qρ ˇpF qρ (4.41)

4.2.4 Linearized Maxwell equations


The linearization of the equations (1.36) are the following:

∇ 3pF qβ `

ˆ

1

2κ´ 2ω

˙

pF qβ “ ´D‹1p ˇpF qρ, ˇpF qσq ` 2 pF qρη, (4.42)

∇ 4pF qβ `

1

2κ pF qβ “ D‹1p ˇpF qρ,´ ˇpF qσq ` 2 pF qρζ (4.43)


∇ 3

(1)

pF qσ ` κ(1)

pF qσ “ 0, (4.44)

∇ 4

(1)

pF qσ ` κ(1)

pF qσ “ 0, (4.45)

∇ 3

(1)

pF qρ` κ(1)

pF qρ “ 0 (4.46)

∇ 4

(1)

pF qρ` κ(1)

pF qρ “ 0 (4.47)

76

and

∇ 3ˇpF qρ` κ ˇpF qρ “ ´

pF qρ`

κ´ κΩ˘

´ div pF qβ (4.48)

∇ 4ˇpF qρ` κ ˇpF qρ “ ´

pF qρκ` div pF qβ (4.49)

∇ 3ˇpF qσ ` κ ˇpF qσ “ curl pF qβ (4.50)

∇ 4ˇpF qσ ` κ ˇpF qσ “ curl pF qβ (4.51)

4.2.5 Linearized Bianchi identities


The linearization of equations (1.38) are the following:

∇ 3α `

ˆ

1

2κ´ 4ω

˙

α “ ´2D‹2 β ´ 3ρpχ´ 2 pF qρ`

D‹2 pF qβ ` pF qρpχ˘

(4.52)

∇ 4α `1

2κα “ 2D‹2 β ´ 3ρpχ` 2 pF qρ

`

D‹2 pF qβ ´ pF qρpχ˘

(4.53)

The linearisation of equations (1.39) and (1.40) are the following:

∇ 3β ` pκ´ 2ωq β “ D‹1p´ρ, σq ` 3ρη

`pF qρ

ˆ

´D‹1p ˇpF qρ, ˇpF qσq ´ κ pF qβ ´1

2κ pF qβ

˙

,(4.54)

∇ 4β ` κβ “ D‹1pρ, σq ` 3ρζ

`pF qρ

ˆ

D‹1p ˇpF qρ,´ ˇpF qσq ´ κ pF qβ ´1

2κ pF qβ

˙

,(4.55)

77

∇ 3β ` p2κ` 2ωq β “ ´div α ´ 3ρξ ` pF qρ`

∇ 3pF qβ ` 2ω pF qβ ` 2 pF qρ ξ

˘

,(4.56)

∇ 4β ` 2κβ “ div α ` pF qρ∇ 4pF qβ (4.57)


∇ 3(1)

ρ`3

2κ

(1)

ρ “ ´2κ pF qρ(1)

pF qρ (4.58)

∇ 4(1)

ρ`3

2κ

(1)

ρ “ ´2κ pF qρ(1)

pF qρ (4.59)

and

∇ 3ρ`3

2κρ “ ´

ˆ

3

2ρ` pF qρ2

˙

pκ´ κΩq ´ 2κ pF qρ ˇpF qρ´ div β ´ pF qρ div pF qβ(4.60)

∇ 4ρ`3

2κρ “ ´

ˆ

3

2ρ` pF qρ2

˙

κ´ 2κ pF qρ ˇpF qρ` div β ` pF qρ div pF qβ (4.61)

∇ 3σ `3

2κσ “ ´ curl β ´ pF qρ curl pF qβ (4.62)

∇ 4σ `3

2κσ “ ´ curl β ´ pF qρ curl pF qβ (4.63)

The above equations (4.7)-(4.63) exhaust all the equations governing the dynam-

ics of linear electromagnetic and gravitational perturbations of Reissner-Nordstrom

spacetime.

78

Chapter 5

Special solutions: pure gauge and

linearized Kerr-Newman

In this chapter, we consider two special linear gravitational and electromagnetic per-

turbations around Reissner-Nordstrom spacetime: the pure gauge solutions and the

linearized Kerr-Newman.

These solutions are of fundamental importance in the proof of linear stability.

The convergence of a linear gravitational and electromagnetic perturbation around

Reissner-Nordstrom spacetime only holds modulo a certain additional gauge freedom

and modulo the convergence to a linearized Kerr-Newman solution.

We describe here in general such solutions and we will specialize in Chapter 8

to the actual choice of gauge and Kerr-Newman parameters in the linear stability.

We begin in Section 5.1 with a discussion of pure gauge solutions to the linearized

Einstein-Maxwell equations, followed by the description of a 6-dimensional family of

linearized Kerr-Newman solutions in Section 5.2.

79

5.1 Pure gauge solutions G

Pure gauge solutions to the linearized Einstein-Maxwell equations are those derived

from linearizing the families of metrics that arise from applying to Reissner-Nordstrom

smooth coordinate transformations which preserve the Bondi form of the metric (2.1).

We will classify such solutions here, making a connection between coordinate trans-

formations and null frame transformations which preserve the Bondi form.

5.1.1 Coordinate and null frame transformations

In order to obtain pure gauge solutions in the setting of linearized Einstein-Maxwell

equations we can equivalently consider coordinate transformations applied to the

metric, or null frame transformations applied to the null frame associated to the

metric. For completeness, we make here a connection between these two approaches.

Coordinate transformations

Consider four functions g1, g2, g3, g4 on the Reissner-Nordstrom manifold, and con-

sider a smooth one-parameter family of coordinates defined by

u “ u` εg1pu, r, θ, φq

r “ r ` εg2pu, r, θ, φq

θ “ θ ` εg3pu, r, θ, φq

φ “ φ` εg4pu, r, θ, φq

If we express the Reissner-Nordstrom metric in the form (3.14)

gM,Q “ ´2dudr ` Ωprqdu2` r2

pdθ2` sin2 θdφ2

q. (5.1)

80

then this defines with respect to the original coordinates u, r, θ, φ a one-parameter

family of metrics. We can classify the coordinate transformations which preserve the

Bondi form of the metric (2.1).

Lemma 5.1.1.1. The general coordinate transformation that preserves the Bondi

form of the metric is given by

u “ u` εg1pu, θ, φq

r “ r ` ε pr ¨ w1pu, θ, φq ` w2pu, θ, φqq

θ “ θ ` ε

ˆ

´1

rpg1qθpu, θ, φq ` j3pu, θ, φq

˙

φ “ φ` ε

ˆ

´1

r sin2 θpg1qφpu, θ, φq ` j4pu, θ, φq

˙

for any function g1pu, θ, φq, w1pu, θ, φq, w2pu, θ, φq, j3pu, θ, φq, j4pu, θ, φq.

Proof. See Section B.1 in the Appendix.

Null frame transformations

Null frame transformations, i.e. linear transformations which take null frames into

null frames, can be thought of as pure gauge transformations, which correspond to a

change of coordinates.

We recall here the classification of null frame transformations.

Lemma 5.1.1.2 (Lemma 2.3.1 in [34]). A general linear null frame transformation

can be written in the form

e14 “ λ`

e4 ` fAeA

˘

,

e13 “ λ´1`

e3 ` fAeA

˘

,

e1A “ OABeB `

1

2fAe4 `

1

2fAe3

81

where λ is a scalar function, f and f are Su,s-tensors and OAB is an orthogonal

transformation of pSu,s, gq, i.e. OACOB

DgCD “ gAB.

Observe that the identity transformation is given by λ “ 1, fA “ fA“ 0 and

OAB“ δBA . Therefore, a linear perturbation of a null frame is a one for which

logpλq “ fA “ fA“ Opεq and OA

B“ δBA `Opεq.

Writing the transformation for the Ricci coefficients and curvature components

under a general null transformation of this type, we have for example (see Proposition

2.3.4. in [34]):

ξ1A “ λ2

ˆ

ξA `1

2λ´1e14pfAq ` ωfA `

1

4κfA

˙

ζ 1A “ ζ ´ e1Aplog λq `1

4p´κfA ` κfAq ` ωfA ´ ωfA

η1A“ η `

1

2λ´1e14pfq `

1

2κf ´ ωf

ω1 “ λ

ˆ

ω ´1

2λ´1e14plog λq

˙

If the metric is in Bondi gauge then it verifies (2.4), i.e. ξA “ 0, ω “ 0 and ηA`ζA “ 0.

This means that a null frame transformation which preserves the Bondi form has to

similarly verify ξ1A “ 0, ω1 “ 0 and η1A` ζ 1A “ 0. This translates into conditions for

e4f , e4f and e4λ. In particular we have the following

Lemma 5.1.1.3. The general null frame transformation that preserves the Bondi

form of the metric is given by a transformation verifying

∇ 4λ “ 0 (5.2)

∇ 4f `1

2κf “ 0 (5.3)

∇ 4f `1

2κf “ 2ωf ´ 2D‹1pλ, 0q (5.4)

82

Proof. Straightforward computation from the above formulas for the change of null

frame.

Relation between coordinate transformations and null frame transforma-

tions

Given a coordinate transformation which preserves the Bondi metric as in Lemma

5.1.1.1, we can associate a null frame transformation between the null frames canon-

ically defined in terms of the vectorfield coordinates by (2.2). In particular, we can

explicitly write the terms which determine the null frame transformation f , f , λ, OAB

in terms of the coordinate transformations g1, w1, w2, jA. We summarize the relation

in the following lemma.

Lemma 5.1.1.4. Given a coordinate transformation which preserves the Bondi form

of the metric as in Lemma 5.1.1.1 of the form



θA “ θA ` ε`

D‹1pg1, 0qpu, θ, φq ` jApu, θ, φq

˘

then the null frame transformation which brings the associated null frame te4, e3, eAu

into te4, e3, eAu is determined by

λ “ 1` εw1

f “ ´εD‹1pg1, 0q

f “ ε p´2rD‹1pw1, 0q ´ 2D‹1pw2, 0q ` ΩprqD‹1pg1, 0qq

OBA “ δBA ` ε

`

∇ A∇ Bg1 `∇ ApjBq˘

83

Proof. See Section B.2 in the Appendix.

Using the above Lemma, the conditions imposed to preserve the Bondi metric in

terms of coordinate transformations or in terms of null frame transformations become

manifest. They are the following:

• The condition for g1 which gives Brpg1q “ 0 translates into

∇ 4f “ ´ε∇ 4D‹1pg1, 0q “ ´εD‹1p∇ 4g1, 0q `1

2κεD‹1pg1, 0q “ ´

1

2κf

which is the condition for the frame coming from imposing ξ “ 0, i.e. (5.3).

• The condition for g2 which gives g2pu, r, θ, φq “ r ¨w1pu, θ, φq`w2pu, θ, φq trans-

lates into

∇ 4f “ ε p´rκD‹1pw1, 0q ´ 2∇ 4D‹1pw2, 0q `∇ 4ΩprqD‹1pg1, 0q ` Ωprq∇ 4D‹1pg1, 0qq

“ ε

ˆ

κD‹1pw2, 0q ´ 2ωD‹1pg1, 0q ´1

2κΩprqD‹1pg1, 0q

˙

Write

εκD‹1pw2, 0q “ ´1

2κf ´ ε2D‹1pw1, 0q ` ε

1


“ ´1

2κf ´ 2D‹1pλ, 0q ` ε

1


and we obtain

∇ 4f “ ´2D‹1pλ, 0q ´1

2κpfq ` 2ωf

which is the condition for the frame coming from imposing η` ζ “ 0, i.e. (5.4).

84

• The condition for g2 which gives g2pu, r, θ, φq “ r ¨ w1pu, θ, φq ` w2pu, θ, φq also

translates into

∇ 4λ “ ∇ 4p1` εw1q “ 0

which is the condition for the frame coming from imposing ω “ 0, i.e. (5.2).

• Writing j “ ´rD‹1pq1, q2q for two functions q1, q2 with vanishing mean, the

conditions for jA which give jA “ jApu, θ, φq translates into

∇ 4q1 “ 0, ∇ 4q2 “ 0

In the next subsections, we will look at the explicit pure gauge solutions produced

by null frame or coordinate transformations preserving the Bondi form of the metric,

and we separate them into

1. pure gauge solutions arising from setting jA “ 0: Lemma 5.1.2.1

2. pure gauge solutions arising from setting log λ “ f “ f “ 0 (or equivalently

g1 “ w1 “ w2 “ 0): Lemma 5.1.3.1

In view of linearity, the general pure gauge solution can be obtained from summing

solutions in the two above cases.

85

5.1.2 Pure gauge solutions with jA “ 0

The following is the explicit form of the pure gauge solution arising from a null

transformation with jA “ 0. Define

h “ ´εg1

h “ ε p´2rw1 ´ 2w2 ` Ωg1q

a “ εw1

then according to Lemma 5.1.1.4, the null frame components can be simplified to

f “ D‹1ph, 0q f “ D‹1ph, 0q λ “ ea

In particular, the relations on f , f and λ given by Lemma 5.1.1.3 translate into

conditions on the derivative along the e4 directions for the functions h, h and a

(conditions (5.5)-(5.7)).

Lemma 5.1.2.1. Let h, h, a be smooth functions, with h, h supported in l ě 1.

Suppose they verify the following transport equations:

∇ 4a “ 0 (5.5)

∇ 4h “ 0 (5.6)

∇ 4h “ 2ωh´ 2a (5.7)

Then the following is a linear gravitational and electromagnetic perturbation around

Reissner-Nordstrom spacetime.

86

The linearized metric components are given by

g “ 2rD‹2D‹1ph, 0q, b “ r∇ 3D‹1ph, 0q ` D‹1ph, 0q,

trγg “ r2p´2rD1D‹1ph, 0q ´ κh´ κhq ,

(1)

Ω “ ´Ωa, Ω “1

2∇ 3h` ωh` Ω

ˆ

1

2∇ 3h´ a

˙

, ς “ ´1

2∇ 3h` a

The Ricci coefficients are given by

pχ “ ´D‹2D‹1ph, 0q, pχ “ ´D‹2D‹1ph, 0q

ζ “

ˆ

´1

4κ´ ω

˙

D‹1ph, 0q `1

4κD‹1ph, 0q ` D‹1pa, 0q,

η “1

2∇ 3D‹1ph, 0q ´ ωD‹1ph, 0q `

1

4κD‹1ph, 0q,

ξ “1

2∇ 3D‹1ph, 0q `

ˆ

1

4κ` ω

˙

D‹1ph, 0q

(1)

κ “ κa,(1)

κ “ ´κa,(1)

ω “1

2∇ 3a´ ωa

κ “ κa` D1D‹1ph, 0q `ˆ

1

4κκ

˙

h`1

4κ2h,

κ “ ´κa` D1D‹1ph, 0q `ˆ

1

4κ2` ωκ

˙

h`

ˆ

1

4κκ´ ρ

˙

h,

ω “1

2∇ 3a´ ωa´

1

2p∇ 3ωqh´

1

2

`

ρ` pF qρ2˘

h,

K “ ´1

4κD1D‹1ph, 0q ´

1

4κD1D‹1ph, 0q `

1

2r2pκh` κhq

The electromagnetic components are given by

pF qβ “pF qρD‹1ph, 0q, pF qβ “ ´ pF qρD‹1ph, 0q

(1)

pF qρ “ 0,(1)

pF qσ “ 0, ˇpF qρ “1

2pF qρ pκh` κhq , ˇpF qσ “ 0

87

The curvature components are given by

α “ 0, α “ 0

β “3

2ρD‹1ph, 0q, β “ ´

3

2ρD‹1ph, 0q

(1)

ρ “ 0, ρ “

ˆ

3

4ρ`

1

2pF qρ2

˙

pκh` κhq , σ “ 0

Proof. We check that the quantities defined above verify the equations in Section 4.2.

We verify some of the equations for metric coefficients. Equations (4.7) and (4.8)

are verified using (5.6) and the fact that κ “ 2r:

∇ 4 g “ ∇ 4p2rD‹2D‹1ph, 0qq “ 2D‹2D‹1ph, 0q ` 2rp´κD‹2D‹1ph, 0qq “ 2pχ

∇ 3 g “ ∇ 3p2rD‹2D‹1ph, 0qq “ rκD‹2D‹1ph, 0q ` 2rpD‹2p∇ 3D‹1ph, 0qq ´1

2κD‹2D‹1ph, 0qq

“ 2pχ` 2D‹2b

Equation (4.11) is verified using (5.6) and the fact that 2ω ´ rρ “ 4ω:

∇ 4b´1

2κb “ ∇ 3D‹1ph, 0q ` r∇ 4∇ 3D‹1ph, 0q `∇ 4D‹1ph, 0q ´

1

2κpr∇ 3D‹1ph, 0q ` D‹1ph, 0qq

“ rpp1

4κκ´ ρqD‹1ph, 0q ´

1

2κ∇ 3D‹1ph, 0qq ` 2ωD‹1ph, 0q

´1

2κD‹1ph, 0q ´ 2D‹1pa, 0q ´

1

2κpD‹1ph, 0qq

“ ´∇ 3D‹1ph, 0q ´ κD‹1ph, 0q `ˆ

1

2κ` 4ω

˙

D‹1ph, 0q ´ 2D‹1pa, 0q “ ´2pη ` ζq

Observe that κ´κΩ “ D1D‹1ph, 0q``

14κ2 ` ωκ

˘

h` 14κκh´ 1

2κ∇ 3h´

12κ∇ 3h. Equations

88

(4.15) and (4.16) are verified:

∇ 4pr´2

trγgq “ ´2D1D‹1ph, 0q ´ 2r∇ 4D1D‹1ph, 0q ´∇ 4κh´∇ 4κh´ κ∇ 4h

“ ´2D1D‹1ph, 0q ´ 2rp´κD1D‹1ph, 0qq ´ p´1

2κκ` 2ρqh

´p´1

2κ2qh´ κp2ωh´ 2aq “ 2κ

∇ 3pr´2

trγgq “ ´rκD1D‹1ph, 0q ´ 2rpD1∇ 3D‹1ph, 0q ´1

2κD1D‹1ph, 0qq

´∇ 3κh´ κ∇ 3h´∇ 3κh´ κ∇ 3h

“ 2pκ´ κΩq ´ 2div b

We verify some of the null structure equations. Equation (4.17) is verified using

(3.34):

∇ 3pχ` pκ` 2ωq pχ` 2D‹2ξ ` α

“ ∇ 3p´D‹2D‹1ph, 0qq ` pκ` 2ωq p´D‹2D‹1ph, 0qq

`2D‹2p1

2∇ 3pD‹1ph, 0qq ` p

1

4κ` ωqD‹1ph, 0qq “ 0

89

Equation (4.21) is verified:

∇ 3ζ `

ˆ

1

2κ´ 2ω

˙

ζ ´ 2D‹1pω, 0q `ˆ

1

2κ` 2ω

˙

η ´1

2κξ ` β ` pF qρ pF qβ

“

ˆ

´1

4p´

1

2κ2´ 2ωκq ´∇ 3ω

˙

D‹1ph, 0q `ˆ

´1

4κ´ ω

˙

∇ 3D‹1ph, 0q

`1

4p´

1

2κκ` 2ωκ` 2ρqD‹1ph, 0q

`1

4κ∇ 3D‹1ph, 0q `∇ 3D‹1pa, 0q

`

ˆ

1

2κ´ 2ω

˙

p

ˆ

´1

4κ´ ω

˙

D‹1ph, 0q `1

4κD‹1ph, 0q ` D‹1pa, 0qq

´2D‹1p1

2∇ 3a´ ωa´

1

2p∇ 3ωqh´

1

2pρ` pF qρ2

qh, 0q

`

ˆ

1

2κ` 2ω

˙

p1

2∇ 3D‹1ph, 0q ´ ωD‹1ph, 0q `

1

4κD‹1ph, 0qq

´1

2κp

1

2∇ 3D‹1ph, 0q `

ˆ

1

4κ` ω

˙

D‹1ph, 0qq ´3

2ρD‹1ph, 0q ` pF qρp´ pF qρD‹1ph, 0qq “ 0

Equation (4.25) is verified, using that D2D‹2 “ 12D‹1D1´K “ 1

2D‹1D1` 1

4κκ`ρ´ pF qρ2:

div pχ`1

2κζ `

1

2D‹1pκ, 0q ´ β ` pF qρ pF qβ

“ ´D2D‹2D‹1ph, 0q `1

2κp

ˆ

´1

4κ´ ω

˙

D‹1ph, 0q `1

4κD‹1ph, 0q ` D‹1pa, 0qq

`1

2D‹1pκ, 0q `

3

2ρD‹1ph, 0q ` pF qρp´ pF qρD‹1ph, 0qq

“ p´1

2D‹1D1 ´

1

8κκ`

1

2ρqD‹1ph, 0q `

ˆ

´1

8κ2´

1

2κω

˙

D‹1ph, 0q `1

2κD‹1pa, 0q

`1

2D‹1p´κa` D1D‹1ph, 0q `

ˆ

1

4κ2` ωκ

˙

h`

ˆ

1

4κκ´ ρ

˙

h, 0q “ 0

Equation (4.27) is verified, using that 2ω ` rρ “ 0:

∇ 3(1)

κ`1

2κ

(1)

κ “ ∇ 3paκq `1

2aκκ “ ∇ 3paqκ “ 2ω

(1)

κ`4

r

(1)

ω ` 2(1)

ρ

90

Equation (4.30) is verified, using (5.5):

∇ 4(1)

κ`1

2κ

(1)

κ “ ∇ 4páκq `1

2κ páκq “ á

ˆ

∇ 4pκq `1

2κκ

˙

“ ´2aρ

ˆ

2M

r2´

2Q2

r3

˙

(1)

κ` 2(1)

ρ “

ˆ

2M

r2´

2Q2

r3

˙

aκ “ ´2aρ

Equation (4.31) is verified, using (5.5):

∇ 4(1)

ω “ ∇ 4páω `1

2∇ 3paqq “ á∇ 4ω “ ápρ`

pF qρ2q

ˆ

M

r2´

3Q2

2r3

˙

(1)

κ “

ˆ

M

r2´

3Q2

2r3

˙

aκ “

ˆ

2M

r3´

3Q2

r4

˙

a “ á

ˆ

´2M

r3`

2Q2

r4`Q2

r4

˙

We verify some of the Maxwell equations. Equation (4.42) is verified:

∇ 3pF qβ `

ˆ

1

2κ´ 2ω

˙

pF qβ ` D‹1p ˇpF qρ, ˇpF qσq ´ 2 pF qρη

“ ´κ pF qρD‹1ph, 0q ` pF qρ∇ 3D‹1ph, 0q

`

ˆ

1

2κ´ 2ω

˙

ppF qρD‹1ph, 0qq ` D‹1p

1

2pF qρ pκh` κhq , 0q

´2 pF qρp1

2∇ 3D‹1ph, 0q ´ ωD‹1ph, 0q `

1

4κD‹1ph, 0qq “ 0


∇ 3ˇpF qρ` κ ˇpF qρ` pF qρ

`

κ´ κΩ˘

` div pF qβ

“ ∇ 3p1

2pF qρ pκh` κhqq ` κp

1

2pF qρ pκh` κhqq

`pF qρ

ˆ

D1D‹1ph, 0q `ˆ

1

4κ2` ωκ

˙

h`1

4κκh´

1

2κ∇ 3h´

1

2κ∇ 3h

˙

´pF qρD1D‹1ph, 0q “ 0

91

We verify some of the Bianchi identities. Equation (4.56) is verified:

∇ 3β ` p2κ` 2ωq β ` div α ` 3ρξ ´ pF qρ`

∇ 3pF qβ ` 2ω pF qβ ` 2 pF qρ ξ

˘

“ ∇ 3p´3

2ρD‹1ph, 0qq ` p2κ` 2ωq p´

3

2ρD‹1ph, 0qq ` 3ρp

1

2∇ 3pD‹1ph, 0qq ` p

1

4κ` ωqD‹1ph, 0qq

´pF qρ

´

∇ 3p´pF qρD‹1ph, 0qq ` 2ωp´ pF qρD‹1ph, 0qq

`2 pF qρ p1

2∇ 3pD‹1ph, 0qq ` p

1

4κ` ωqD‹1ph, 0qq

¯

“ ´3

2p´

3

2κρ´ κ pF qρ2

qD‹1ph, 0q ` p2κq p´3

2ρD‹1ph, 0qq ` 3ρpp

1

4κqD‹1ph, 0qq

´pF qρ

ˆ

´p´κ pF qρqD‹1ph, 0q ` 2 pF qρ pp1

4κqD‹1ph, 0qq

˙

“ 0

Equation (4.61) is verified, using (5.6) and (5.7):

∇ 4ρ`3

2κρ`

ˆ

3

2ρ` pF qρ2

˙

κ` 2κ pF qρ ˇpF qρ´ div β ´ pF qρdiv pF qβ

“

ˆ

´3

2ρκκ´ 2 pF qρ2κκ`

3

2ρ2` ρ pF qρ2

˙

h`

ˆ

´3

2ρκ2

´ 2 pF qρ2κ2

˙

h

`

ˆ

3

4ρκ`

1

2pF qρ2κ

˙

p2ωh´ 2aq `3

2κp

ˆ

3

4ρ`

1

2pF qρ2

˙

κh`

ˆ

3

4ρ`

1

2pF qρ2

˙

κhq

`

ˆ

3

2ρ` pF qρ2

˙

pκa` D1D‹1ph, 0q `ˆ

1

4κκ´ ωκ´ ρ

˙

h`1

4κ2hq

`2κ pF qρ1

2pF qρ pκh` κhq ´ div p

3

2ρD‹1ph, 0qq ´ pF qρ div p pF qρD‹1ph, 0qq “ 0

The remaining equations are verified in a similar manner.

5.1.3 Pure gauge solutions with g1 “ w1 “ w2 “ 0

The following is the explicit form of the pure gauge solution arising from a null

transformation for which e3 and e4 are unchanged, while the frame on the spheres

change. They only generate non-trivial values for the metric components, while all

other quantities of the solution vanish.

92

Define

j “ ´rD‹1pq1, q2q

for two functions q1, q2 with vanishing mean.

Lemma 5.1.3.1. Let q1, q2 be smooth functions, with q1 supported in l ě 1 and q2

supported in l ě 2 spherical harmonics.1 Suppose they verify the following transport

equations:

∇ 4q1 “ 0 (5.8)

∇ 4q2 “ 0 (5.9)

Then the following is a linear gravitational and electromagnetic perturbation around

Reissner-Nordstrom spacetime. The linearized metric components are given by

g “ 2r2D‹2D‹1pq1, q2q, trγg “ ´2r4D1D‹1pq1, 0q, b “ r2D‹1p∇ 3q1,∇ 3q2q

while all other components of the solution vanish.

Proof. Equations (4.7) and (4.8) are verified using (5.8) and (5.8):

∇ 4 g “ 2∇ 4pr2D‹2D‹1pq1, q2qq “ 2r2D‹2D‹1p∇ 4q1,∇ 4q2q “ 0

∇ 3 g “ 2∇ 3pr2D‹2D‹1pq1, q2qq “ 2r2D‹2D‹1p∇ 3q1,∇ 3q2q “ 2D‹2b

1The pure gauge solution corresponding to q1 “ 0 and q2 “ 9Y l“1m generates the trivial solution.

93


∇ 4b´1

2κb “ ∇ 4pr

2D‹1p∇ 3q1,∇ 3q2qq ´1

2κr2D‹1p∇ 3q1,∇ 3q2q

“ rD‹1p∇ 3q1,∇ 3q2q ` rD‹1p∇ 4∇ 3q1,∇ 4∇ 3q2q ´1

2κr2D‹1p∇ 3q1,∇ 3q2q “ 0

Equations (4.15) and (4.16) are verified:

∇ 4pr´2

trγgq “ ∇ 4

`

´2r2D1D‹1pq1, 0q˘

“`

´2r2D1D‹1p∇ 4q1, 0q˘

“ 0

∇ 3pr´2

trγgq “ ∇ 3

`

´2r2D1D‹1pq1, 0q˘

“`

´2r2D1D‹1p∇ 3q1, 0q˘

“ ´2div b

All the other equations are trivially satisfied.

We identify the solutions given by Lemma 5.1.2.1 and Lemma 5.1.3.1 to pure

gauge solutions, and we denote their linear sum as Gph,h,a,q1,q2q.

5.1.4 Gauge-invariant quantities

We can identify quantities which vanish for any gauge transformation Gph,h,a,q1,q2q.

Such quantities are referred to as gauge-invariant.

The symmetric traceless two tensors α, α are clearly gauge-invariant from Lemma

5.1.2.1. These curvature components are important because in the case of the Einstein

vacuum equation they verify a decoupled wave equation, the celebrated Teukolsky

equation, first discovered in the Schwarzschild case in [5] and generalized to the Kerr

case in [44]. In the Einstein-Maxwell case, the tensors α and α verify Teukolsky

equations coupled with new quantities, denoted f and f.

The symmetric traceless 2-tensors

f :“ D‹2 pF qβ ` pF qρpχ and f :“ D‹2 pF qβ ´ pF qρpχ (5.10)

94

are gauge-invariant quantities. Indeed, using Lemma 5.1.2.1, we see that for every

gauge solution

f “ D‹2 pF qβ ` pF qρpχ “ D‹2p pF qρD‹1ph, 0qq ` pF qρp´D‹2D‹1ph, 0qq “ 0

Similarly for f.

Notice that the quantities f and f appear in the Bianchi identities for α and α.

The equations (4.52) and (4.53) can be rewritten as

∇ 3α `

ˆ

1

2κ´ 4ω

˙

α “ ´2D‹2 β ´ 3ρpχ´ 2 pF qρ f, (5.11)

∇ 4α `

ˆ

1

2κ´ 4ω

˙

α “ 2D‹2 β ´ 3ρpχ` 2 pF qρ f (5.12)

Using the above, it is clear that f and f shall appear on the right hand side of the

wave equation verified by α and α.

The quantities f and f themselves verify Teukolsky-type equations, which are cou-

pled with α and α respectively. The equations for α and f and for α and f constitute

the generalized spin ˘2 Teukolsky system obtained in Section 6.1.

Observe that the extreme electromagnetic component pF qβ and pF qβ are not gauge-

invariant if pF qρ is not zero in the background.2 On the other hand, the one-forms

β :“ 2 pF qρβ ´ 3ρ pF qβ and β :“ 2 pF qρβ ´ 3ρ pF qβ (5.13)

are gauge invariant. Indeed, using Lemma 5.1.2.1, we see that for every gauge solution

β “ 2 pF qρβ ´ 3ρ pF qβ “ 2 pF qρp3

2ρD‹1ph, 0qq ´ 3ρp pF qρD‹1ph, 0qq “ 0

2In the case of the Maxwell equations in Schwarzschild, the components pF qβ and pF qβ are gauge-invariant, and satisfy a spin ˘1 Teukolsky equation, see [39].

95

and similarly for β.

5.2 A 6-dimensional linearised Kerr-Newman fam-

ily K

The other class of special solutions corresponds to the family that arises by lineariz-

ing one-parameter representations of Kerr-Newman around Reissner-Nordstrom. We

will present such a family here, giving first in Section 5.2.1 a 3-dimensional family

corresponding to Kerr-Newman with fixed angular momentum a (supported in l “ 0

spherical harmonics) and then in Section 5.2.2, a 3-dimensional family corresponding

to Kerr-Newman with fixed mass M and charge Q (supported in l “ 1 spherical

harmonics).

5.2.1 Linearized Kerr-Newman solutions with no angular mo-

mentum

Reissner-Nordstrom spacetimes are obviously solutions to the nonlinear Einstein-

Maxwell equation. Therefore, linearization around the parameters M and Q give

rise to solution of the linearized system of gravitational and electromagnetic pertur-

bations, which can be interpreted as the solution converging to another Reissner-

Nordstrom solution with a small change in the mass or in the charge.

In addition to those, there is a family of solutions with non-trivial magnetic charge,

which can arise as solution of the Einstein-Maxwell equations. Indeed, the following

expression gives stationary solutions to the Maxwell system on Reissner-Nordstrom:

F “b

r2εAB `

Q

r2dt^ dr

96

where b and Q are two real parameters, respectively the magnetic and the electric

charge.

We summarize these solutions in the following Proposition.

Proposition 5.2.1.1. For every M,Q, b P R, the following is a (spherically symmet-

ric) solution of the system of gravitational and electromagnetic perturbations in M.

The non-vanishing quantities are

(1)

ρ “

ˆ

´2M

r3`

4QQ

r4

˙

,(1)

pF qρ “Q

r2

(1)

pF qσ “b

r2

(1)

κ “

ˆ

4M

r2´

4QQ

r3

˙

,(1)

ω “

ˆ

M

r2´

2QQ

r3

˙

,(1)

Ω “

ˆ

2M

r´

2QQ

r2

˙

Proof. We verify the equations in Section 4.2 which are not trivially satisfied.


(1)

κ “

ˆ

4M

r2´

4QQ

r3

˙

“2

r

ˆ

2M

r´

2QQ

r2

˙

“ κ(1)

Ω

Equation (4.27) is given by

4

r

(1)

ω ` 2(1)

ρ “4

r

ˆ

m

r2´

2QQ

r3

˙

` 2

ˆ

´2m

r3`

4QQ

r4

˙

“ 0


∇ 3(1)

κ`1

2κ

(1)

κ “ ∇ 3

ˆ

4M

r2´

4QQ

r3

˙

`1

2κ

ˆ

4M

r2´

4QQ

r3

˙

“

ˆ

´8M

r2`

12QQ

r3

˙

1

2κ`

1

2κ

ˆ

4M

r2´

4QQ

r3

˙

“

ˆ

´4M

r2`

8QQ

r3

˙

1

2κ “ ´2κ

(1)

ω

97


∇ 4(1)

κ`1

2κ

(1)

κ “ ∇ 4

ˆ

4M

r2´

4QQ

r3

˙

`1

2κ

ˆ

4M

r2´

4QQ

r3

˙

“

ˆ

´4M

r3`

8QQ

r4

˙

“ 2(1)

ρ

Equation (4.31) reads:

∇ 4(1)

ω “ ∇ 4

ˆ

m

r2´

2QQ

r3

˙

“

ˆ

´2m

r2`

6QQ

r3

˙

“(1)

ρ`2Q

r2

(1)

pF qρ


´1

4κ

(1)

κ´1

4κ

(1)

κ´(1)

ρ` 2 pF qρ(1)

pF qρ “ ´1

4

2

r

ˆ

4m

r2´

4QQ

r3

˙

´

ˆ

´2m

r3`

4QQ

r4

˙

` 2Q

r2

Q

r2“ 0

The Maxwell equations (4.44)-(4.45) and (4.46)-(4.47) are verified:

∇ 3

(1)

pF qσ ` κ(1)

pF qσ “ ∇ 3pb

r2q ` κ

b

r2“ ´2

b

r3∇ 3r ` κ

b

r2“ ´2

b

r2

1

2κ` κ

b

r2“ 0

∇ 3

(1)

pF qρ` κ(1)

pF qρ “ ∇ 3pQ

r2q ` κ

Q

r2“ ´2

Q

r3∇ 3r ` κ

Q

r2“ ´2

Q

r2

1

2κ` κ

Q

r2“ 0

The Bianchi identities (4.58)-(4.59) are verified:

∇ 3(1)

ρ`3

2κ

(1)

ρ`2Q

r2κ

(1)

pF qρ “ ∇ 3p

ˆ

´2m

r3`

4QQ

r4

˙

q `3

2κ

ˆ

´2m

r3`

4QQ

r4

˙

`2Q

r2κQ

r2

“

ˆ

6m

r4´

16QQ

r5

˙

∇ 3prq ` κ

ˆ

´3m

r3`

8QQ

r4

˙

“

ˆ

6m

r3´

16QQ

r4

˙

1

2κ` κ

ˆ

´3m

r3`

8QQ

r4

˙

“ 0

This proves the proposition.

98

5.2.2 Linearized Kerr-Newman solutions leaving the mass

and the charge unchanged

In addition to the variation of mass and change in the Reissner-Nordstrom solution,

a variation in the angular momentum is also possible. We therefore have to take into

account the perturbation into a Kerr-Newman solution for small a.

We start from the Kerr-Newman metric expressed in outgoing Eddington-Finkelstein

coordinates ignoring all terms quadratic or higher in a:

gKŃ “ ´2drdu´

ˆ

1´2M

r`Q2

r2

˙

du2` r2

`

dθ2` sin2 θdφ2

˘

´

ˆ

4Mr ´ 2Q2

r2

˙

a sin2 θdudφ` 2a sin2 θdrdφ

Notice that these coordinates do not realize the Bondi gauge, because of the presence

of the last term pdrdφq, which is not allowed in the Bondi form (2.1).

Performing the change of coordinates

φ1 “ φà

r

we obtain, ignoring all terms quadratic in a:

gKŃ “ ´2drdu´

ˆ

1´2M

r`Q2

r2

˙

du2` r2

´

dθ2` sin2 θpdφ1q2 ´ 2 sin2 θ

a

r2drdφ1

¯

´

ˆ

4Mr ´ 2Q2

r2

˙

a sin2 θdudφ1 ` 2a sin2 θdrdφ1

“ ´2drdu´

ˆ

1´2M

r`Q2

r2

˙

du2` r2

`

dθ2` sin2 θpdφ1q2

˘

´

ˆ

4Mr ´ 2Q2

r2

˙

a sin2 θdudφ1

which is the Kerr-Newman metric in Bondi gauge, and clearly of the form (4.1) with

99

a “ εa. Linearizing in a, we can read off the linearized metric coefficients, and notice

that the only one non-vanishing is b. We obtain the following solutions:

Proposition 5.2.2.1. Let 9Y `“1m for m “ ´1, 0, 1 denote the l “ 1 spherical harmonics

on the unit sphere. For any a P R, the following is a smooth solution of the system

of gravitational and electromagnetic perturbations on M.

The only non-vanishing metric coefficient is

b “

ˆ

´8M

r`

4Q2

r2

˙

aεABBB 9Y `“1m

The only non-vanishing Ricci coefficient is

ζ “ η “

ˆ

´6M

r2`

4Q2

r3

˙

aεABBB 9Y `“1m

The non-vanishing electromagnetic components are

pF qβ “Q

rκaεABBB 9Y `“1

m , pF qβ “Q

rκaεABBB 9Y `“1

m , ˇpF qσ “ ´4Q

r3a ¨ 9Y `“1

m

The non-vanishing curvature components are

β “

ˆ

´3M

r2`

3Q2

r3

˙

κaεABBB 9Y `“1m , β “

ˆ

´3M

r2`

3Q2

r3

˙

κaεABBB 9Y `“1m

σ “

ˆ

´12M

r4`

8Q2

r5

˙

a 9Y `“1m

Note that the above family may be parametrised by the ` “ 1-modes of the

electromagnetic component ˇpF qσ.

Proof. We verify the equations in Section 4.2. Since D‹2b, div b “ 0, the only non-trivial

100

equation for the metric coefficients is (4.11), which is verified:

∇ 4b´1

2κb “ ∇ 4

´

ˆ

´8M

r`

4Q2

r2

˙

aεABBB 9Y `“1m

¯

´1

2κ

ˆ

´8M

r`

4Q2

r2

˙

aεABBB 9Y `“1m

“

ˆ

8M

r2´

8Q2

r3

˙

aεABBB 9Y `“1m ´ κ

ˆ

´8M

r`

4Q2

r2

˙

aεABBB 9Y `“1m

“

ˆ

24M

r2´

16Q2

r3

˙

aεABBB 9Y `“1m “ ´2pη ` ζq

Since D‹2ζ, div ζ, D‹2η, div η “ 0, the only non-trivial null structure equations are

(4.21)-(4.22), the Codazzi equations (4.25)-(4.26) and (4.39). To verify (4.21) and

(4.22), recalling that ζ “ η, we compute

∇ 3ζ ` κζ “ ∇ 3p

ˆ

´6M

r2`

4Q2

r3

˙

aεABBB 9Y `“1m q ` κ

ˆ

´6M

r2`

4Q2

r3

˙

aεABBB 9Y `“1m

“

ˆ

6M

r2´

6Q2

r3

˙

κaεABBB 9Y `“1m `

1

2κ

ˆ

´6M

r2`

4Q2

r3

˙

aεABBB 9Y `“1m

“

ˆ

3M

r2´

4Q2

r3

˙

κaεABBB 9Y `“1m “ ´β ´ pF qρ pF qβ

To verify Codazzi equations (4.25)-(4.26), we compute

´1

2κζ ´

1

2D‹1pκ, 0q ` β ´ pF qρ pF qβ

“ ´1

2κ

ˆ

´6M

r2`

4Q2

r3

˙

aεABBB 9Y `“1m `

ˆ

´3M

r2`

3Q2

r3

˙

κaεABBB 9Y `“1m

´Q2

r3κaεABBB 9Y `“1

m “ 0

To verify (4.39), we compute:

curl ζ “ εBABBζ “

ˆ

´6M

r2`

4Q2

r3

˙

aεBABBεACBC 9Y `“1

m “

ˆ

6M

r4´

4Q2

r5

˙

a4 S29Y `“1m

“

ˆ

´12M

r4`

8Q2

r5

˙

a 9Y `“1m “ σ

101

Since div pF qβ, div pF qβ “ 0, the Maxwell equations (4.44)-(4.49) are trivially satis-

fied. To verify (4.42) and (4.43), we compute

∇ 4pF qβ `

1

2κ pF qβ “ ∇ 4

ˆ

Q

rκaεABBB 9Y `“1

m

˙

`1

2κ

ˆ

Q

rκaεABBB 9Y `“1

m

˙

“ ´Q

2rκκaεABBB 9Y `“1

m `Q

r

ˆ

´1

2κκ` 2ρ

˙

aεABBB 9Y `“1m

“Q

r3

ˆ

4´12M

r`

8Q2

r2

˙

aεABBB 9Y `“1m

D‹1p ˇpF qρ,´ ˇpF qσq ` 2 pF qρζ “4Q

r3aεABBB 9Y `“1

m ` 2Q

r2

ˆ

´6M

r2`

4Q2

r3

˙

aεABBB 9Y `“1m

“Q

r3

ˆ

4´12M

r`

8Q2

r2

˙

aεABBB 9Y `“1m

To verify (4.50)-(4.51), we compute:

∇ 3ˇpF qσ ` κ ˇpF qσ “ ∇ 3

ˆ

´4Q

r3

˙

a ¨ 9Y `“1m ´ κ

4Q

r3a ¨ 9Y `“1

m “2Q

r3aκ ¨ 9Y `“1

m

curl pF qβ “ ´Q

r3κa4 S2

9Y `“1m “

2Q

r3κa 9Y `“1

m

Since D‹2β, D‹2β “ 0 and div β, div β “ 0, the Bianchi identities (4.52)-(4.53) and

(4.60)-(4.61) are trivially satisfied. To verify (4.57), we compute

∇ 4β ` 2κβ “ ∇ 4p

ˆ

´3M

r2`

3Q2

r3

˙

κaεABBB 9Y `“1m q ` 2κ

ˆ

´3M

r2`

3Q2

r3

˙

κaεABBB 9Y `“1m

“

´

ˆ

3M

r2´

92Q2

r3

˙

κ2`

ˆ

´3M

r2`

3Q2

r3

˙

p´κ2q

`2κ2

ˆ

´3M

r2`

3Q2

r3

˙

¯

aεABBB 9Y `“1m “

ˆ

´32Q2

r3

˙

κ2aεABBB 9Y `“1m

pF qρ∇ 4pF qβ “

Q

r2∇ 4

ˆ

Q

rκaεABBB 9Y `“1

m

˙

“ ´Q

r2

Q

2rκ2aεABBB 9Y `“1

m `Q

r2

Q

rp´κ2

qaεABBB 9Y `“1m

“

ˆ

´32Q2

r3

˙

κ2aεABBB 9Y `“1m

102

and similarly for (4.56). To verify (4.55) we compute:

∇ 4β ` κβ “ ∇ 4p

ˆ

´3M

r2`

3Q2

r3

˙

κaεABBB 9Y `“1m q ` κ

ˆ

´3M

r2`

3Q2

r3

˙

κaεABBB 9Y `“1m

“

ˆ

3M

r2´

92Q2

r3

˙

κκaεABBB 9Y `“1m `

ˆ

´3M

r2`

3Q2

r3

˙

p´1

2κκ` 2ρqaεABBB 9Y `“1

m

`

ˆ

´3M

r2`

3Q2

r3

˙

p´1

2κκqaεABBB 9Y `“1

m ` κ

ˆ

´3M

r2`

3Q2

r3

˙

κaεABBB 9Y `“1m

“

ˆ

3M

r2´

92Q2

r3

˙

κκaεABBB 9Y `“1m `

ˆ

´3M

r2`

3Q2

r3

˙

2ρaεABBB 9Y `“1m

“

´

´12M

r4`

36M2

r5`

18Q2

r5´

72MQ2

r6`

30Q4

r7

¯

aεABBB 9Y `“1m

while the right hand side is given by:

D‹1pρ, σq ` 3ζρ` pF qρ

ˆ

D‹1p ˇpF qρ,´ ˇpF qσq ´ κ pF qβ ´1

2κ pF qβ

˙

“

“

ˆ

´12M

r4`

8Q2

r5

˙

aεABBB 9Y `“1m ` 3ρ

ˆ

´6M

r2`

4Q2

r3

˙

aεABBB 9Y `“1m

`pF qρ

ˆ

4Q

r3aεABBB 9Y `“1

m ´ κQ

rκaεABBB 9Y `“1

m ´1

2κQ

rκaεABBB 9Y `“1

m

˙

“

ˆ

´12M

r4`

8Q2

r5

˙

aεABBB 9Y `“1m ` 3ρ

ˆ

´6M

r2`

4Q2

r3

˙

aεABBB 9Y `“1m

`pF qρ

ˆ

4Q

r3aεABBB 9Y `“1

m ´3

2

Q

rκκaεABBB 9Y `“1

m

˙

“

´

ˆ

´12M

r4`

8Q2

r5

˙

` 3

ˆ

´2M

r3`

2Q2

r4

˙ˆ

´6M

r2`

4Q2

r3

˙

`Q

r2p4Q

r3´

3

2

Q

rκκq

¯

aεABBB 9Y `“1m

“

´

´12M

r4`

36M2

r5`

18Q2

r5´

72MQ2

r6`

30Q4

r7

¯

aεABBB 9Y `“1m

103

To verify (4.62) and (4.63), we compute

∇ 3σ `3

2κσ “ ∇ 3

ˆ

´12M

r4`

8Q2

r5

˙

a 9Y `“1m `

3

2κ

ˆ

´12M

r4`

8Q2

r5

˙

a 9Y `“1m

“

ˆ

6M

r4´

8Q2

r5

˙

κa 9Y `“1m

´ curl β ´ pF qρ curl pF qβ “

ˆ

6M

r4´

6Q2

r5

˙

κa 9Y `“1m ´

2Q2

r5κa 9Y `“1

m

which proves the proposition.

Observe that the above Proposition describes for each a P R and for each m “

´1, 0, 1 a solution to the linearized Einstein-Maxwell equations corresponding to a

linearized Kerr-Newman solution. The reason why we have a 3-dimensional family

of solution varying the angular momentum is identical to the case of Schwarzschild.

Indeed, linearizing the metric at a non-trivial (a ‰ 0) member of the Kerr-Newman

family creates non-trivial pure gauge solutions corresponding to a rotation of the axis.

On the other hand, while linearizing the spherically symmetric Reissner-Nordstrom

metric these rotations correspond to trivial pure gauge solutions. The 3-dimensional

family above corresponds then to the identification of the axis of symmetry (a unit

vector in R3) and the angular momentum of the solution (the length of the vector).

We combine the 3-dimensional space of solutions of Proposition 5.2.1.1 and the

3-dimensional space of solutions of Proposition 5.2.2.1 in the following definition:

Definition 5.2.1. Let M, Q, b, a´1, a0, a1 be six real parameters. We call the sum

of the solution of Proposition 5.2.1.1 with parameters m, Q and b and the solution

of Proposition 5.2.2.1 satisfying ˇpF qσ “ř

amY`“1m the linearized Kerr-Newman

solution with parameters pM,Q, b, a´1, a0, a1q and denote it by KpM,Q,b,aq or simply

K .

104

Remark 5.2.1. Observe that the gauge-invariant quantities β and β defined by (5.13)

have the additional remarkable property that they vanish for every linearised Kerr-

Newman solution Kpm,Q,b,aq. Indeed, for every a,

β “ 2 pF qρβ ´ 3ρ pF qβ

“ 2Q

r2

ˆ

´3M

r2`

3Q2

r3

˙

κaεABBBY`“1m ´ 3

ˆ

´2M

r3`

2Q2

r4

˙

Q

rκaεABBBY

`“1m “ 0

105

Chapter 6

The Teukolsky equations and the

decay for the gauge-invariant

quantities

In this chapter, we will introduce the generalized Teukolsky equations of spin ˘2 and

the generalized Teukolsky equation of spin ˘1 which govern the gravitational and

electromagnetic perturbations of Reissner-Nordstrom spacetime. We also introduce

the generalized Regge-Wheeler system and the generalized Fackerell-Ipser equation

and the connection between the two sets of equations, through the Chandrasekhar

transformation.

These equations have a fundamental relation to the problem of linear stability for

gravitational and electromagnetic perturbations of Reissner-Nordstrom spacetime.

Indeed, the gauge invariant quantities identified in the previous chapter verify the

generalized Teukolsky equations. The Chandrasekhar transformation shall allow us

to derive estimates for these quantities.

In Section 6.1 we define the generalized spin ˘2 Teukolsky equations and the

106

generalized Regge-Wheeler system considering them as a second order hyperbolic

PDEs for independent unknowns α, α, f, f and q, qF.

In Section 6.2 we define the generalized spin ˘1 Teukolsky equations and the

generalized Fackerell-Ipser equation considering them as a second order hyperbolic

PDEs for independent unknowns β, β and p.

In Section 6.3 we introduce a fundamental transformation mapping solutions of

the generalized Teukolsky equation to solutions to the Regge-Wheeler/Fackerell-Ipser

equations. This transformation plays an important role in deriving estimates for this

equation. The Chandrasekhar transformation here defined generalizes the physical

space definition given in [16] to the case of Reissner-Nordstrom, and identifies one

operator which is applied to all quantities involved.

In Section 6.4, we explain the relation of the above PDEs with the full system

of linear gravitational and electromagnetic perturbations of Reissner-Nordstrom. As

guessed from the notation, the curvature and electromagnetic components α, α, f, f,

β, β verify the Teukolsky equations respectively.

Finally in Section 6.5 we state the main theorems in [27] and [28] which give

control and quantitative decay statements of the gauge invariant quantities α, α, f, f,

β, β. We state here the control in L2 and L8 norms which are needed in the following

for the proof of linear stability in Chapter 9. We also recall the main ideas of the

proofs.

The main results on this chapter have appeared in [27] and [28].

107

6.1 The spin ˘2 Teukolsky equations and the Regge-

Wheeler system

In this section, we will introduce a generalization of the celebrated spin ˘2 Teukolsky

equations and the Regge-Wheeler equation, and explain the connection between them.

6.1.1 Generalized spin ˘2 Teukolsky system

The generalized spin ˘2 Teukolsky system concern symmetric traceless 2-tensors in

Reissner-Nordstrom spacetime, which we denote pα, fq and pα, fq respectively.

Definition 6.1.1. Let α and f be two symmetric traceless 2-tensors defined on a

subset D ĂM. We say that pα, fq satisfy the generalized Teukolsky system of

spin `2 if they satisfy the following coupled system of PDEs:

lgα “ ´4ω∇ 4α ` 2 pκ` 2ωq∇ 3α

`

ˆ

1

2κκ´ 4ρ` 4 pF qρ2

` 2ω κ´ 10ωκ´ 8ωω ´ 4∇ 4ω

˙

α

`4 pF qρ p∇ 4f` pκ` 2ωq fq ,

lgprfq “ ´2ω∇ 4prfq ` pκ` 2ωq∇ 3prfq `

ˆ

´1

2κκ´ 3ρ` ωκ´ 3ωκ´ 2∇ 4ω

˙

rf

´r pF qρ p∇ 3α ` pκ´ 4ωqαq

where lg “ gµνDµDν denotes the wave operator in Reissner-Nordstrom spacetime.

Let α and f be two symmetric traceless 2-tensor defined on a subset D ĂM. We

say that pα, fq satisfy the generalized Teukolsky system of spin ´2 if they satisfy

108

the following coupled system of PDEs:

lgα “ ´4ω∇ 3α ` 2 pκ` 2ωq∇ 4α

`

ˆ

1

2κκ´ 4ρ` 4 pF qρ2

` 2ω κ´ 10ωκ´ 8ωω ´ 4∇ 3ω

˙

α

´4 pF qρ`

∇ 3f` pκ` 2ωq f˘

,

lgprfq “ ´2ω∇ 3prfq ` pκ` 2ωq∇ 4prfq `

ˆ

´1

2κκ´ 3ρ` ωκ´ 3ωκ´ 2∇ 3ω

˙

rf

`r pF qρ p∇ 4α ` pκ´ 4ωqαq

We note that the generalized Teukolsky system of spin ´2 is obtained from that

of spin `2 by interchanging ∇ 3 with ∇ 4 and underline quantities with non-underlined

ones.

Remark 6.1.1. When the electromagnetic tensor vanishes, i.e. if pF qβ “ pF qβ “

pF qρ “ pF qσ “ 0, the generalized Teukolsky system of spin ˘2 reduces to the first

equation, since f “ 0. Moreover, the first equation reduces to the Teukolsky equation

of spin ˘2 in Schwarzschild.

6.1.2 Generalized Regge-Wheeler system

The other generalized system to be defined here is the generalized Regge-Wheeler

system, to be satisfied again by symmetric traceless tensors pq, qFq.

Definition 6.1.2. Let q and qF be two symmetric traceless 2-tensors on D. We say

that pq, qFq satisfy the generalized Regge–Wheeler system for spin `2 if they

109

satisfy the following coupled system of PDEs:

lgq``

κκ´ 10 pF qρ2˘

q “ pF qρ

˜

4r4 2qF´ 4rκ∇ 4pq

Fq ´ 4rκ∇ 3pq

Fq

` r`

6κκ` 16ρ` 8 pF qρ2˘

qF

¸

`pF qρpl.o.t.q1,

lgqF` pκκ` 3ρq qF “ pF qρ

˜

´1

rq

¸

`pF qρ2

pl.o.t.q2

(6.1)

where pl.o.t.q1 and pl.o.t.q2 are lower order terms with respect to q and qF. Schemat-

ically ∇ ď23 pl.o.tq “ q.

Let q and qF be two symmetric traceless 2-tensors on D. We say that pq, qFq

satisfy the generalized Regge–Wheeler system for spin ´2 if they satisfy the

following coupled system of PDEs:

lgq``

κκ´ 10 pF qρ2˘

q “ ´ pF qρ

˜

4r4 2qF´ 4rκ∇ 4pq

Fq ´ 4rκ∇ 3pq

Fq

` r`

6κκ` 16ρ` 8 pF qρ2˘

qF

¸

´pF qρpl.o.t.q1,

lgqF` pκκ` 3ρq qF “ pF qρ

˜

1

rq

¸

`pF qρ2

pl.o.t.q2

(6.2)

where pl.o.t.q1 and pl.o.t.q2 are lower order terms with respect to q and qF. Schemat-

ically ∇ ď23 pl.o.tq “ q.

In Section 6.3, we will show that given a solution pα, fq and pα, fq of the spin

˘2 Teukolsky equations, respectively, we can derive two solutions pq, qFq and pq, qFq,

respectively, of the generalized Regge-Wheeler system.

Standard well-posedness results hold for both the generalized Teukolsky system

of spin ˘2 and the generalized Regge-Wheeler system.

110

6.2 The spin ˘1 Teukolsky equation and the Fackerell-

Ipser equation

In this section, we introduce a generalization of the celebrated spin ˘1 Teukolsky

equations and the Fackerell-Ipser equation, and explain the connection between them.

6.2.1 Generalized spin ˘1 Teukolsky equation

The generalized spin ˘1 Teukolsky equation concerns 1-tensors which we denote β

and β respectively.

Definition 6.2.1. Let β be a 1-tensor defined on a subset D ĂM. We say that β

satisfy the generalized Teukolsky equation of spin `1 if it satisfies the following

PDE:

lgpr3βq “ ´2ω∇ 4pr

3βq ` pκ` 2ωq∇ 3pr3βq

`

ˆ

1

4κκ´ 3ωκ` ωκ´ 2ρ` 3 pF qρ2

´ 8ωω ` 2∇ 3ω

˙

r3β

´2r3κ pF qρ2

ˆ

∇ 4pF qβ ` p

3

2κ` 2ωq pF qβ ´ 2 pF qρξ

˙

` I

where lg “ gµνDµDν denotes the wave operator in Reissner-Nordstrom spacetime,

and I is a 1-tensor with vanishing projection to the l “ 1 spherical harmonics, i.e.

div Il“1 “ curl Il“1 “ 0.

Let β be a 1-tensor defined on a subset D Ă M. We say that β satisfy the

111

generalized Teukolsky equation of spin ´1 if it satisfies the following PDE:

lgpr3βq “ ´2ω∇ 3pr

3βq ` pκ` 2ωq∇ 4pr3βq

`

ˆ

1

4κκ´ 3ωκ` ωκ´ 2ρ` 3 pF qρ2

´ 8ωω ` 2∇ 4ω

˙

r3β

´2r3κ pF qρ2

ˆ

∇ 3pF qβ ` p

3

2κ` 2ωq pF qβ ` 2 pF qρξ

˙

` I

where I is a 1-tensor with vanishing projection to the l “ 1 spherical harmonics.

Remark 6.2.1. When the electromagnetic tensor of the background vanishes, i.e.

if pF qρ “ 0, the generalized Teukolsky equation of spin ˘1 reduces to the standard

Teukolsky equation of spin ˘1 verified by the extreme components of the electromag-

netic component in Schwarzschild.

6.2.2 Generalized Fackerell-Ipser equation in l “ 1 mode

The other generalized equation in l “ 1 to be defined here is the generalized Fackerell-

Ipser equation, to be satisfied by a one tensor p.

Definition 6.2.2. Let p be a 1-tensor on D Ă M. We say that p satisfies the

generalized Fackerell-Ipser equation in l “ 1 if it satisfies the following PDE:

lgp`

ˆ

1

4κκ´ 5 pF qρ2

˙

p “ J (6.3)

where J is a 1-tensor with vanishing projection to the l “ 1 spherical harmonics, i.e.

div Jl“1 “ curl Jl“1 “ 0.

In Section 6.3, we will show that given a solution β and β of the generalized spin

˘1 Teukolsky equations in l “ 1, respectively, we can derive two solutions p and p,

respectively, of the generalized Fackerell-Ipser equation in l “ 1.

112

Standard well-posedness results hold for both the generalized Teukolsky equation

of spin ˘1 and the generalized Fackerell-Ipser equation in l “ 1.

6.3 The Chandrasekhar transformation

We now describe a transformation theory relating solutions of the generalized Teukol-

sky equations defined above to solutions of the generalized Regge-Wheeler system or

the Fackerell-Ipser equation. We emphasize that a physical space version of the Chan-

drasekhar transformation was first introduced in [16], for the Schwarzschild spacetime.

We introduce the following operators for a n-rank S-tensor Ψ:

P pΨq “1

κ∇ 3prΨq, P pΨq “

1

κ∇ 4prΨq (6.4)

Observe that the operators P and P above preserve the signature of the tensor Ψ as

well as its rank. These operators can be thought of as rescaled derivatives: P is a

rescaled version of ∇ 3, while P is a rescaled version of ∇ 4.

Given a solution pα, fq of the generalized Teukolsky system of spin `2 and a solu-

tion β of the generalized Teukolsky equation of spin `1, we can define the following

113

derived quantities for pα, fq and β:

ψ0 “ r2κ2α,

ψ1 “ P pψ0q,

ψ2 “ P pψ1q “ P pP pψ0qq “: q,

ψ3 “ r2κ f,

ψ4 “ P pψ3q “: qF

ψ5 “ r4κ β,

ψ6 “ P pψ5q :“ p

(6.5)

Similarly, given a solution pα, fq of the generalized Teukolsky system of spin ´2 and

given a solution β of the generalized Teukolsky equation of spin ´1, we can define

the following derived quantities for pα, fq and β:

ψ0“ r2κ2α,

ψ1“ P pψ

0q,

ψ2“ P pψ

1q “ P pP pψ

0qq “: q,

ψ3“ r2κ f,

ψ4“ P pψ

3q “: qF

ψ5“ r4κ β,

ψ6“ P pψ

5q :“ p

(6.6)

These quantities are again symmetric traceless S 2-tensors.

Remark 6.3.1. Observe that, even if f is a symmetric traceless 2-tensor, we apply

the Chandrasekhar transformation only once to obtain the quantity qF which verifies

114

a Regge-Wheeler-type equation (as opposed to α, for which the Chandrasekhar trans-

formation is applied twice). This is because f by definition is constructed from the

one-form pF qβ, which verifies a spin `1 Teukolsky-type equation.

The following proposition is proved in [27] and [28].

Proposition 6.3.0.1. Let pα, fq be a solution of the generalized Teukolsky system of

spin `2. Then the symmetric traceless tensors pq, qFq as defined through (6.5) satisfy

the generalized Regge-Wheeler system of spin `2. Similarly, let pα, fq be a solution

of the generalized Teukolsky system of spin ´2. Then the symmetric traceless tensors

pq, qFq as defined through (6.6) satisfy the generalized Regge-Wheeler system of spin

´2.

Let β be a solution of the generalized Teukolsky equation of spin `1. Then the

1-tensor p as defined through (6.5) satisfies the generalized Fackerell-Ipser equation in

l “ 1. Similarly, let β be a solution of the generalized Teukolsky equation of spin ´1.

Then the 1-tensor p as defined through (6.6) satisfies the generalized Fackerell-Ipser

equation in l “ 1.

6.4 Relation to the gravitational and electromag-

netic perturbations of Reissner-Nordstrom space-

time

We finally relate the equations presented above to the full system of linearized grav-

itational and electromagnetic perturbations of Reissner-Nordstrom spacetime in the

context of linear stability of Reissner-Nordstrom.

The relation is summarized in the following Theorem, proved in [27] and [28].

115

Theorem 6.4.1. Let α, α, f, f, β, β be the curvature components of a linear gravita-

tional and electromagnetic perturbation around Reissner-Nordstrom spacetime as in

Definition 4.2.1.

Then

• pα, fq satisfy the generalized Teukolsky system of spin `2, and pα, fq satisfy the

generalized Teukolsky system of spin ´2.

• β satisfies the generalized Teukolsky equation of spin `1, and β satisfies the

generalized Teukolsky equation of spin ´1.

Using Proposition 6.3.0.1, we can therefore associate to any solution to the lin-

earized Einstein-Maxwell equations around Reissner-Nordstrom spacetime, two sym-

metric traceless 2-tensors which verify the generalized Regge-Wheeler system of spin

˘2 and a one form which verifies the generalized Fackerell-Ipser equation in l “ 1.

6.4.1 Gravitational versus electromagnetic radiation

The linear perturbations considered in Definition 4.2.1 allow the perturbation of the

Weyl curvature of the spacetime, as well as the perturbation of the Ricci curvature, in

the form of the electromagnetic tensor. We will refer to the perturbation of the Weyl

tensor W as gravitational radiation, and to the perturbation of the electromagnetic

tensor F as electromagnetic radiation.

The radiation is always transported by the gauge-invariant versions of the extreme

component of the tensor, i.e. the tensor α represents the gravitational radiation, and

the tensor β represents the electromagnetic radiation.

The combined gravitational and electromagnetic perturbations of Reissner-Nordstrom

spacetime is not solved by simply considering the separated gravitational and electro-

magnetic perturbations. In fact, as already clear from the Bianchi identites and the

116

Maxwell equations above, the interaction between these two perturbations is complex,

and the two radiations are very much coupled together.

The gravitational radiation will still be transported by α, but this gauge-independent

quantity will verify a generalized Teukolsky equation which is coupled to another

gauge independent quantity, f, which has contributions from the electromagnetic ten-

sor. Being both symmetric traceless 2-tensors, they are in general supported in l ě 2

spherical harmonics. Therefore, these two tensors, coming from both the Weyl and

the Ricci perturbations, are responsible for the gravitational radiation in l ě 2.

The quantity pF qβ is not gauge-invariant in the presence of the Weyl perturbation.

It is β, defined using both the curvature and the electromagnetic tensor, to be gauge

invariant and to transport the electromagnetic perturbation. Since it is a 1-form, it is

in general supported in l ě 1 spherical harmonics, which is where the electromagnetic

radiation is supported.

In summary, the gravitational and electromagnetic perturbations of Reissner-

Nordstrom are just, by effect of the above equations, totally coupled together.

6.5 Boundedness and decay for the gauge-invariant

quantities

The main results in [27] and [28] are the boundedness and quantitative decay state-

ments obtained for the gauge-invariant quantities α, α, f, f, β, β verifying the above

Teukolsky equations. Using Theorem 6.4.1, we can summarize the results in the

following.

We denote A À B if A ď CB where C is an universal constant depending on

117

appropriate Sobolev norms of initial data. We define the following norms:

||f ||8pu, rq :“ ||f ||L8pSu,rq

||f ||8,kpu, rq :“kÿ

i“0

||dif ||L8pSu,rq

where d “ t∇ 3, r∇ 4, r∇ u.

Theorem 6.5.1. [Main Theorem in [27] and Main Theorem in [28]] Let α, α, f,

f, β, β be the curvature components of a linear gravitational and electromagnetic

perturbation around Reissner-Nordstrom spacetime. Then for every k we have:

1. The following energy estimates hold true:

ż

Σpτq

r2`δ|α|2 ` r4`δ

|∇ 4α|2` r4`δ

|∇α|2 ` r2`δ|∇ 3α|

2

ďpinitial data for α, f, ψ1, q, qFq

u2´2δż

Σpτq

r2`δ|f|2 ` r4`δ

|∇ 4f|2` r4`δ

|∇ f|2 ` r2`δ|∇ 3f|

2

ďpinitial data for α, f, ψ1, q, qFq

u2´2δż

Σpτq

r6`δ|β|2 ` r8`δ

|∇ 4β|2` r8`δ

|∇ β|2 ` r6`δ|∇ 3β|

2

ďpinitial data for α, f, ψ1, β, q, qF, pq

u2´2δ

(6.7)

and

ż

Σpτq

|α|2 ` r2|∇ 4α|

2` r2

|∇α|2 ` |∇ 3α|2ďpinitial data for α, f, ψ

1, q, qFq

u2´2δ

ż

Σpτq

r2|f|2 ` r4

|∇ 4f|2` r4

|∇ f|2 ` r2|∇ 3f|

2ďpinitial data for α, f, ψ

1, q, qFq

u2´2δ

ż

Σpτq

r6|β|2 ` r8

|∇ 4β|2` r8

|∇ β|2 ` r6|∇ 3β|

2ďpinitial data for α, f, ψ

1, β, q, qF, pq

u2´2δ

(6.8)

118

2. The following pointwise estimates for q, qF and p hold true:

||q||8,k À mintr´1u´12`δ, u´1`δu (6.9)

||qF||8,k À mintr´1u´12`δ, u´1`δu (6.10)

||p||8,k À mintr´1u´12`δ, u´1`δu (6.11)

3. The following pointwise estimates for α, f and β hold true:

||α||8,k À mintr´3´δu´12`δ, r´2´δu´1`δu (6.12)

||f||8,k À mintr´3´δu´12`δ, r´2´δu´1`δu (6.13)

||β||8,k À mintr´5´δu´12`δ, r´4´δu´1`δu (6.14)

4. The following pointwise estimates for α, f and β hold true:

||α||8,k À r´1u´1`δ (6.15)

||f||8,k À r´2u´1`δ (6.16)

||β||8,k À r´4u´1`δ (6.17)

6.5.1 Proof of Theorem 6.5.1

We recall here the main ideas and steps of the proof of Theorem 6.5.1.

Estimates for the Teukolsky system of spin ˘2

According to Theorem 6.4.1, the curvature components pα, fq and pα, fq satisfy the

generalized Teukolsky system of spin ˘2, and according to Proposition 6.3.0.1 we

can associate to them pairs of tensors pq, qFq and pq, qFq which verify the generalized

119

Regge-Wheeler system (6.1) and (6.2). We obtain estimates for these pairs of tensors.

We write system (6.1) in the following concise form:

$

’

’

&

’

’

%

´

lg ´ V1

¯

q “ M1rq, qFs :“ Q C1rq

Fs `Q L1rqFs `Q2 L1rqs,

´

lg ´ V2

¯

qF “ M2rq, qFs :“ Q C2rqs `Q

2L2rqFs

where

C1rqFs “

4

r4 2q

F´

4

rκ∇ 4q

F´

4

rκ∇ 3q

F`

1

r

`

6κκ` 16ρ` 8 pF qρ2˘

qF,

C2rqs “ ´1

r3q,

L1rqs “ ´2

r2ψ0 ´

4

r3ψ1,

L1rqFs “ ´12ρψ3 ´Q

2 40

r4ψ3,

L2rqFs “

4

r3ψ3

and |Q| !M is the charge of the Reissner-Nordstrom spacetime pM, gM,Qq.

The terms Cs and Ls are respectively the coupling and the lower order terms. In

particular:

• The terms C1 and C2 are the terms representing the coupling between the Weyl

curvature and the Ricci curvature. In the wave equation for q the coupling term

C1 “ C1rqFs is an expression in terms of qF, while in the wave equation for qF

the coupling term C2 “ C2rqs is an expression in terms of q.

• The terms L1 and L2 collect the lower order terms: in particular L1rqs are lower

order terms with respect to q, while L1rqFs and L2rq

Fs are lower terms with

respect to qF. The index 1 or 2 denotes if they appear in the first or in the

second equation.

120

As observed in the case of the Regge-Wheeler-type equation obtained in Kerr

in [17], the complete decoupling of the equation is not necessary in the derivation of

the estimates. A new important feature appearing in the Einstein-Maxwell equations,

which is not present in the vacuum case, are the estimates involving coupling terms of

curvature and electromagnetic tensor, which are independent quantities. In addition

to those, the coupling of the Regge-Wheeler equations involve lower order terms, as

in Kerr ([17] and [36]). In order to take into account this whole structure in the

estimates, the two equations have to be considered as one system, together with

transport estimates for the lower order terms.

Estimates for the Regge-Wheeler equations separately

We first derive separated estimates for the two equations composing the system (6.1)

of the form

´

lg ´ Vi

¯

Ψi “ Mi (6.18)

where Mi are whatever expressions we have on the left hand side of the equation.

The two equations comprising the system are obtained by

Ψ1 “ q, q, V1 “ ´κκ` 10 pF qρ2“

1

r2

ˆ

4´8M

r`

14Q2

r2

˙

, (6.19)

Ψ2 “ qF, qF V2 “ ´κκ´ 3ρ “1

r2

ˆ

4´2M

r´

2Q2

r2

˙

(6.20)

We apply to both equations separately the standard procedures used to derive

energy-Morawetz estimates. We use standard techniques in order to derive the esti-

mates: we apply the vectorfield method to obtain energy and Morawetz estimates.

We also improve the estimates with the red-shift vector field. Finally, we use the

121

rp method of Dafermos and Rodnianski to obtain integrated decay in the far-away

region. We also obtain higher order estimates by commuting the equations with the

Killing vector fields.

We give here an overview of the vectorfield method used in this context.

Consider the energy-momentum tensor associated to the wave equation (6.18):

Qµν : “ DµΨ ¨DνΨ´1

2gµν

`

DλΨ ¨DλΨ` ViΨ ¨Ψ

˘

“ DµΨ ¨DνΨ´1

2gµνLirΨs

(6.21)

Let X “ aprqe3 ` bprqe4 be a vectorfield, w a scalar function and M a one form.

Defining

PpX,w,Mqµ rΨs “ QµνXν`

1

2wΨDµΨ´

1

4Ψ2Bµw `

1

4Ψ2Mµ,

then a standard computation shows that

DµPpX,w,Mqµ rΨs “1

2Q ¨ pXqπ `

ˆ

´1

2XpViq ´

1

4lgw

˙

|Ψ|2 `1

2wLirΨs `

1

4DµpΨ2Mµq

`

ˆ

XpΨq `1

2wΨ

˙

¨MirΨs

(6.22)

Defining

ErX,w,M spΨq :“ DµPpX,w,Mqµ rΨs ´

ˆ

XpΨq `1

2wΨ

˙

¨Mi (6.23)

then equation (6.22) becomes

ErX,w,M spΨq “ 1

2Q ¨ pXqπ `

ˆ

´1

2XpViq ´

1

4lgw

˙

|Ψ|2

`1

2wLirΨs `

1

49DµpΨ2Mµq

(6.24)

122

The estimates are obtained by applying the divergence theorem to relation (6.23)

to a casual portion of the spacetime, for the following choice of vectorfields:

• X “ Bt: energy estimates

• X “ fprqBr (for a well-chosen radial function f): Morawetz estimates

• X “ rpe4: rp-hierarchy estimates

The vectorfields are chosen so that the bulk termsş

M E and the boundary termsş

ΣP ¨n are positive definite, and they constitute the spacetime bulk energies and the

energies of the solution respectively. Define the energy and the Morawetz bulks as

EprΨspτq : “

ż

Στ

|∇ 4Ψ|2 ` |∇ 3Ψ|2 ` |∇Ψ|2 ` r´2|Ψ|2 `

ż

ΣěRpτq

rp|∇ 4Ψ|2

MprΨspτ1, τ2q : “

ż

Mpτ1,τ2q

|RpΨq|2 ` |TΨ|2 ` |∇Ψ|2 ` |Ψ|2

`

ż

MěRpτ1,τ2q

rp´1´

p|∇ 4pΨq|2` p2´ pqp|∇Ψ|2 ` r´2

|Ψ|2q¯

By application of the divergence theorem we then obtain

EprΨspτq `MprΨsp0, τq À EprΨsp0q

´

ż

Mp0,τq

ˆ

pr ´ rP qRpΨq ` T pΨq `1

2wΨ

˙

¨MirΨs(6.25)

which can be applied to both equations for q and qF.

Notice that these separated estimates contain on the right hand side terms in-

volving M1 and M2 that at this stage are not controlled. In particular M1 and M2

contain both the coupling terms C and the lower order terms L.

We observe that the structure of the right hand side in the two equations of

the system is not symmetric. In particular, the coupling term C1rqFs in the first

123

equation involves up to two derivatives of qF, while the coupling term C2rqs in the

second equation contains 0th-order derivative of q. In order to take into account the

difference in the presence of derivatives, we consider the 0th-order Morawetz and rp

weighted estimate for the first equation of the form (6.25) and the 1st-order estimate

for the second equation, obtained by commuting the equations with the Killing vector

fields Bt and angular derivatives. We add those two estimates together. This operation

will create a combined estimate, where the Morawetz bulks on the left hand side of

each equations shall absorb the coupling term on the right hand side of the other

equation.

Estimates for the coupling and lower order terms

We derive estimates for the coupling terms on the right hand side and transport

estimates for the lower order terms on the right hand side. Our goal is to absorb

the norms of these inhomogeneous terms on the right hand side with the Morawetz

bulks of the estimates on the left hand side, using the smallness of the charge. More

precisely, we shall absorb the integrals

´

ż

Mp0,τq

ˆ

pr ´ rP qRpqq ` T pqq `1

2wq

˙

¨M1rq, qFs

´

ż

Mp0,τq

ˆ

pr ´ rP qTRpqFq ` TT pqFq `

1

2wTqF

˙

¨ TM2rq, qFs

´

ż

Mp0,τq

ˆ

pr ´ rP q∇RpqFq `∇T pqFq `1

2w∇ qF

˙

¨∇M2rq, qFs

by the Morawetz bulks

Mprqsp0, τq `M1,T,∇p rqFsp0, τq

124

The Morawetz bulks Mprqsp0, τq contain the Br derivative and the zero-th order

term, i.e. |Rq|2 and |q|2, and the Bt and the angular derivative with degeneracy at

the trapping region, i.e. pr ´ rP q2p|Tq|2 ` |∇ q|2q, where rP is the photon sphere of

Reissner-Nordstrom. Similarly M1,T,∇p rqFsp0, τq contains the above commuted with

T and ∇ .

Outside the trapping region, the absorption of the integrals on the right hand

side into the Morawetz bulks on the left hand side are straightforward using Cauchy-

Schwarz. For example, considering one of the highest order terms:

ż

Mp0,τqzr“rp

Rpqq ¨4 qF ď

˜

ż

Mp0,τqzr“rp

|Rpqq|2

¸12 ˜ż

Mp0,τqzr“rp

|4 qF|2¸12

À

ż

Mp0,τqzr“rp

|Rpqq|2 `

ż

Mp0,τqzr“rp

|4 qF|2

À Mprqsp0, τq `M1,T,∇p rqFsp0, τq

Since this integral is multiplied by the charge Q, the smallness of the charge allows

for the absorbtion of these integrals in the left hand side.

In the trapping region, this absorption is delicate because of the degeneracy of

the bulk norms. Some terms can still be bounded by Cauchy-Schwarz, being careful

to distribute the degeneracy to the correct terms. For example

´

ż

Mtrap

ppr ´ rP qRpqqq ¨4

r4 2q

Fď

˜

ż

Mtrap

|Rpqq|2

¸12 ˜ż

Mtrap

pr ´ rP q2|4 2q

F|2

¸12

À Mprqsp0, τq `M1,T,∇p rqFsp0, τq

where Mtrap indicates the a spacetime neighborhood of tr “ rP u.

On the other hand, terms involving only T and angular derivatives (both of which

125

are degenerate in the bulks) cannot be absorbed in this way. The following terms

´

ż

Mtrap

T pqq ¨4

r4 2q

F´

ż

Mtrap

T pTqFq ¨ T

ˆ

´1

r3q

˙

´

ż

Mtrap

T pr∇ AqFq ¨ r∇ Aˆ

´1

r3q

˙

cannot be absorbed as in the previous way.

Neverthless, the special structure of the coupling terms C1rqFs and C2rqs implies

a cancellation of these problematic terms in the trapping region. To allow for a

cancellation, we multiply the estimates above by positive constants A, B, C. Upon

performing integration by parts and using the wave equations the above three terms

can be brought in terms which have the same structure:

´A

ż

Mtrap

T pqq ¨4

r4 2q

F´B

ż

Mtrap

Υ4 2qF¨ T

ˆ

´1

r3q

˙

´ C

ż

Mtrap

pr24 2qFq ¨

ˆ

´1

r3Tq

˙

Choosing the constant A, B and C such that

4Ar2´BΥprq ´ Cr2

|r“rP “ 0,

`

4Ar2´BΥprq ´ Cr2

˘1|r“rP “ p8Ar ´BΥ1

prq ´ 2Crq|r“rP “ 0

(6.26)

we obtain a cancellation of second order for the terms involving Tq ¨ 4 2qF at the

photon sphere.

Observe that a choice of positive constants A, B, C verifying conditions (6.26) is

possible. Indeed, since ΥprP q ě 0 and Υ1prP q ě 0, there exists a choice of positive

constants A, B, C such that 4Ar2P ´BΥprP q ´ Cr

2p “ 8ArP ´BΥ1prP q ´ 2CrP “ 0.

Observe that this cancellation is possible because of two reasons: the higher order

terms in the coupling terms C1rqFs (i.e. 4

r4 2q

F) and C2rqs (i.e. ´ 1r3q) have opposite

sign, and the higher order term is second order derivatives. We emphasize that the

particular structure of the coupling terms on the right hand side allows the estimates

126

to be derived as in [27].

The lower order terms are treated using enhanced transport estimates, which

make also use of Bianchi identities. To absorb the lower order terms L1rqs, L1rqFs

and L2rqFs in the combined estimate, we derive transport estimates for α and f using

the differential relations (6.5), to get non-degenerate energy estimates. Using these

estimates, we will be able to control the norms involving the lower terms.

Summing the separated estimates and absorbing the coupling terms and the lower

order terms on the right hand side we obtain a combined estimate for the system as

in the Main Theorem in [27]:

Eprqspτq ` E1,T,∇p rqFspτq `Mprqsp0, τq `M1,T,∇

p rqFsp0, τq

À Eprqsp0q ` E1,T,∇p rqFsp0q ` Erfsp0q ` Eprψ1sp0q ` Eprαsp0q

(6.27)

and

Eprαspτq ` Eprψ1spτq ` E1,T,∇p rfspτq

À Eprqsp0q ` E1,T,∇p rqFsp0q ` Eprαsp0q ` Eprψ1sp0q ` E

1,T,∇p rfsp0q

(6.28)

and higher order derivative estimates.

Pointwise estimates for α, f, α, f follow by the rp hierarchy estimates and standard

Sobolev embedding.

Estimates for the Teukolsky equation of spin ˘1

According to Theorem 6.4.1, the curvature components β and β satisfy the generalized

Teukolsky equation of spin ˘1, and according to Proposition 6.3.0.1 we can associate

to them p and p which verify the generalized Fackerell-Ipser equation of spin ˘1 (6.3).

More precisely, the derived quantity p verifies the following generalized Fackerell-Ipser

127

equation in l “ 1:

lgp`

ˆ

1

4κκ´ 5 pF qρ2

˙

p “ 8r2 pF qρ2div pqFq (6.29)

which is of the form given in Definition 6.2.2, with J “ 8r2 pF qρ2div pqFq, which has

vanishing projection to the l “ 1 mode. Observe that the right hand side is already

controlled by the previous steps. We can therefore derive estimates for the one-form

p by standard techniques applied to the above equation, with controlled right hand

side.

More precisely, applying the vectorfield method as described above to the Fackerell-

Ipser equation we obtain

Eprpspτq `Mprpsp0, τq À Eprpsp0q ´Q2

ż

Mp0,τq

ˆ

pr ´ rP qRppq ` T ppq `1

2wp

˙

¨ div qF(6.30)

As explained above, the absorption of the integral on the right hand side outside

the trapping region can be done using Cauchy-Schwarz and the estimate (6.27). For

example

ż

Mp0,τqzr“rp

Rppq ¨ div qF ď

˜

ż

Mp0,τqzr“rp

|Rppq|2

¸12 ˜ż

Mp0,τqzr“rp

|∇ qF|2¸12

À

ż

Mp0,τqzr“rp

|Rppq|2 `

ż

Mp0,τqzr“rp

|∇ qF|2

À Mprpsp0, τq `M1,T,∇p rqFsp0, τq

À Mprpsp0, τq ` Eprqsp0q ` E1,T,∇p rqFsp0q

`Erfsp0q ` Eprψ1sp0q ` Eprαsp0q

The first term on the last line, which is multiplied by the charge, can be absorbed

by the same term on the left hand side of (6.30). Similarly, the integral involving

128

pr ´ rP qRppq ¨ div qF can be absorbed using Cauchy-Schwarz.

On the other hand, the term involving the T derivative can be simplified using

the relations between p, q and qF:

D‹2p “ ´r pF qρq´ r2p3ρ` 2 pF qρ2

qqF ` 4r5 pF qρ2κf (6.31)

Indeed, upon integration by partsş

MtrapTp ¨ div qF simplifies to

ż

Mtrap

D‹2p ¨ TqF “ż

Mtrap

p´r pF qρq´ r2p3ρ` 2 pF qρ2

qqF ` 4r5 pF qρ2κfq ¨ TqF

which can easily be bounded by the Morawetz bulks of q, qF and f, and therefore by

initial data. This will yield

Eprpspτq `Mprpsp0, τq À Eprpsp0q ` Eprqsp0q ` E1,T,∇p rqFsp0q

` Erfsp0q ` Eprψ1sp0q ` Eprαsp0q

(6.32)

Using transport estimates we can then obtain estimates for β:

Eprβspτq À Eprpsp0q ` Eprβsp0q ` Eprqsp0q ` E1,T,∇p rqFsp0q

` Eprαsp0q ` Eprψ1sp0q ` E1,T,∇p rfsp0q

(6.33)

Alternatively, we can project equation (6.29) to the l “ 1 spherical harmonics

and estimate the projection to the l “ 1 spherical mode of p, which, together with

transport estimates, will give control on the l “ 1 spherical mode of β and β. Observe

that the quantities α, f and β are related by the following relation:

r3κD‹2β “ ´pF qρψ1 ´

`

2 pF qρ2` 3ρ

˘

r3κf

129

By these relations, it is clear that the bounds and decay obtained for ψ1 and f in the

Main Theorem in [27] imply bounds and decay for D‹2β, therefore on the projection

to the l ě 2 spherical harmonics of β. Using the control for the l “ 1 mode of β

and β obtained through the generalized Fackerell-Ipser equation in l “ 1 and elliptic

estimates, we can derive control for the one-tensors β and β. Pointwise estimate for

β and β are then obtained by standard Sobolev inequalities.

130

Chapter 7

Initial data and well-posedness

In this chapter, we consider the well-posedness of the system of linearized gravitational

and electromagnetic perturbations.

We first describe how to prescribe initial data in Section 7.1 and we define what

it means for data to be asymptotically flat in Section 7.2. Finally we formulate the

well-posedness theorem in Section 7.3.

7.1 Seed data on an initial cone

We describe here how to prescribe initial data for the linearized Einstein-Maxwell

equations of Section 4.2.

We present a characteristic initial value problem. We fix a sphere S0 :“ Su0,r0

in M, obtained as intersection of two hypersurfaces for some values tr “ r0u and

tu “ u0u. Consider the outgoing Reissner-Nordstrom light cone C0 :“ Cu0 , and the

ingoing Reissner-Nordstrom light cone C0 on which the data are being prescribed.

Initial data are prescribed by so-called seed data that can be prescribed freely.

Definition 7.1.1. Given a sphere S0 with corresponding null cones C0 and C0, a

131

smooth seed initial data set consists of prescribing

• along C0: a smooth symmetric traceless 2-tensor g0,outand a smooth 1-form

pF qβ0,

• along C0: a smooth symmetric traceless 2-tensor g0,in, which coincides with g0,in

on S0,

• along C0: smooth 1-forms b0, ξ0, and pF qβ

0

• along C0: smooth functions Ω0, ω0,(1)

Ω0,(1)

ω0

• on the sphere S0: a smooth 1-form ζ0,

• on the sphere S0: smooth functions trγg0, κ0, κ0, ˇpF qρ0, ˇpF qσ0,

(1)

κ0,(1)

pF qρ0,(1)

pF qσ0.

We will show in Theorem 7.3.1 that the above freely prescribed tensors uniquely

determine a solution to the linear gravitational and electromagnetic perturbation of

Reissner-Nordstrom.

7.2 Asymptotic flatness of initial data

We first define the following derived quantities along C0 from a smooth seed initial

data as in Definition 7.1.1:

pχ0,out “1

2∇ 4 g0,out

α0,out “ ´1

2r´2∇ 4pr

2∇ 4 g0,outq

Note that these quantities are uniquely determined in terms of the seed data.

132

For a tensor ξ we define for n1 ě 0, n2 ě 0:

Dn1,n2ξ “ pr∇ qn1pr∇ 4qn2ξ

We define the following notion of asymptotic flatness of initial data.

Definition 7.2.1. We call a seed data set asymptotically flat with weight s to order

n if the seed data satisfies the following estimates along C0 for some 0 ă s ď 1 and

any n1 ě 0, n2 ě 0 with n1 ` n2 ď n:

|Dn1,n2pr2pχ0,outq| ` |Dn1,n2pr

3`sα0,outq| ` |Dn1,n2pr2`s pF qβ0q| ď C0,n1,n2 (7.1)

for some constant C0,n1,n2 depending on n1 and n2.

We will show in Theorem 7.3.1 that asymptotically flat seed data lead in particular

to a hierarchy of decay for all quantities on the initial data.

7.3 The well-posedness theorem

We can now state the fundamental well-posedness theorem for linear gravitational

and electromagnetic perturbations of Reissner-Nordstrom.

Theorem 7.3.1. Fix a sphere S0 and consider a smooth seed initial data set as in

Definition 7.1.1. Then there exists a unique smooth solution S of linear gravitational

and electromagnetic perturbations around Reissner-Nordstrom spacetime defined in

MX I`pS0q which agrees with the seed data on C0 and C0.

Moreover, suppose the smooth seed initial data set is asymptotically flat with weight

133

s to order n. Then on the initial cone C0, the following estimates hold:

|Dkpr3`sαq| ` |Dkpr3`sβq| ` |Dkpr3ρq| ` |Dkpr3σq| ` |Dkpr2βq| ` |Dkprαq| ď C

|Dkpr2`s pF qβq| ` |Dkpr2 ˇpF qρq| ` |Dkpr2 ˇpF qσq| ` |Dkpr pF qβq| ď C

|Dkpr2pχq| ` |Dkprpχq| ` |Dkpr2ζq| ` |Dkprηq| ` |Dkprξq| ` |Dkpr2κq| ` |Dkprκq| ď C

for any k ď n´ 3 and a constant which can be computed explicitly from initial data.

Proof. We first show that the equations uniquely determine from seed data all dy-

namical quantitites on C0 Y C0 such that all tangential equations are satisfied.

We first note that the seed data determines on the initial sphere S0:

• (1)

ρ from (4.37)

• σ from (4.39)

• K from (4.5)

• ρ from (4.41)

• β from (4.25)

• β from (4.26)

• η from (4.10)

• ς from (4.9)

We now integrate our seed data from S0 along the cone C0.

•(1)

pF qρ and(1)

pF qσ are determined by integrating (4.46) and (4.44) along C0, and(1)

ρ

is determined integrating (4.58).

134

• (1)

κ is determined integrating (4.29) and(1)

κ is determined by (4.27).

• The tensors g and b0 are part of the seed data on C0. Note that these determine

uniquely pχ via (4.8).

• The tensor ξ is part of the seed data, therefore α is determined using (4.17).

• From (4.35) we have an ODE for κ with prescribed right hand side along C0

and the value of κ is prescribed at S0. Therefore it uniquely determines κ along

C0.

• From (4.16) the value of trγg is determined, and therefore using (4.5) K is

determined.

• Integrating (4.48) and (4.50) we see that ˇpF qρ and ˇpF qσ are determined.

• From (4.56) we have an ODE for β which determines it along C0.

• Integrating (4.60) and (4.62) we see that ρ and σ are determined.

• By (4.41), κ is determined.

• Using (4.25), ζ is determined.

• From (4.10), η is determined.

• From (4.9), ς is determined.

• Integrating (4.42) from S0 we obtain that pF qβ is determined along C0.

• Integrating (4.19) we obtain that pχ is determined.

• By (4.26) we have that β is determined.

• Using (4.52) we finally obtain that α is determined.

135

We now integrate our seed data from S0 along the cone C0. We prove simultane-

ously that the seed data determines the solution on the cone, and also the estimates

in the case of asymptotically flat seed initial data.

• Recall that the seed data g uniquely determines pχ and α by (4.7) and (4.18).

• From (4.32), we have ∇ 4pr2κq “ 0 which gives |r2κ| ď C where C can be

computed explicitly from the seed data.

• From (4.49), we have ∇ 4pr2 ˇpF qρq “ ´Qκ`r2div pF qβ. We see that the right hand

side is integrable by the asymptotic flatness condition, therefore this produces

the bound |r2 ˇpF qρ| ď C. Similarly for ˇpF qσ from (4.51).

• From (4.57) and the asymptotic flatness condition we obtain the uniform bound

|r3`sβ| ď C.

• From (4.22) and the asymptotic flatness condition we obtain |r2ζ| ď C.

• From (4.43) we obtain |r pF qβ| ď C.

• From (4.61) and (4.63) we obtain |r3ρ| ` |r3σ| ď C.

• From (4.55) we obtain |r2β| ď C.

• Finally, from (4.24), (4.20), (4.34), (4.23), (4.53) we obtain the bounds for η, pχ,

κ, ξ and α.

By Theorem 6.4.1, given a solution S , the quantities α, α, f, f satisfy the gener-

alized Teukolsky system of spin ˘2. Therefore, using the well-posedness property of

the Teukolsky system, we can determine globally those quantities from their initial

values on C0 Y C0.

136

Similarly, by Theorem 6.4.1, given a solution S , the quantities β, β satisfy the

generalized Teukolsky equation of spin ˘1. Therefore, using the well-posedness prop-

erty of the Teukolsky system, we can determine globally those quantities from their

initial values on C0 Y C0.

Once these are determined, we will determine all the remaining quantities by

integrating transport equations or by taking derivatives.

For example, given α, equation (4.18) can be integrated as a linear o.d.e. from

seed data in C0 to determine pχ. Given f “ D‹2 pF qβ` ρpχ, then pF qβ can be determined

modulo its projection to the l “ 1 mode. Then from β “ 2 pF qρβ ´ 3ρ pF qβ, β is

determined for l ě 2 mode. From (4.32), κ is determined by seed data. Therefore,

knowing pF qβ, equation (4.49) can be integrated to determine ˇpF qρ. Similarly, equation

(4.61) can be integrated to determine ρ. Integrating (4.36), we can determine ω from

the initial data seed, and integrating (4.14) we determine Ω. Using Codazzi equation

(4.26), we can determine ζ for l ě 2 modes. We may continue ordering the remaining

equations hierarchically so all reimaining quantities are determined by the previous

by integrating transport equations or by taking derivatives.

137

Chapter 8

Gauge-normalized solutions and

identification of the Kerr-Newman

parameters

In this chapter we define the gauge normalizations that will play a fundamental role

in the proof of linear stability. We also identify the correct Kerr-Newman parameters

of a solution to the linearized Einstein-Maxwell equations.

We first define in Section 8.1 what it means for a solution S to be initial data

normalized. Such a solution will be used to prove that any solution is bounded,

upon a choice of a gauge solution which can be expressed in terms of initial data.

Moreover, this choice of gauge implies decay for most components of the solution,

but it leads to an incomplete result in terms of decay of some of the components.

To overcome this difficulty, in Section 8.2 we define what it means for a solution S

to be SU,R-normalized. Such normalization takes place at a sphere far-away in the

spacetime, and is reminiscent of the choice of gauge at the last slice in [34]. This new

normalization will be used in the next chapter to obtain the complete optimal decay

138

for all the components of a solution S .

We then show in Section 8.3 and Section 8.4 that given a solution S of the

linearized Einstein-Maxwell equations we can indeed associate to it an initial data

normalized solution, and if S is bounded we can associate a SU,R-normalized solution.

They are respectively denoted S id and S U,R, and are realized by adding to S a pure

gauge solution G of the form described in Section 5.1.

The pure gauge solution used to obtain the initial data normalization is explicitly

computable from initial data. On the other hand, the one used to obtain the SU,R-

normalization is not. Only in the proof of the decay in the next chapter, in Section

9.2, we will show that the pure gauge solution used to achieve the SU,R-normalization

is itself bounded by initial data.

Finally, in Section 8.5 we identify the Kerr-Newman parameters out of the pro-

jection of the solution to the l “ 0, 1 modes of the initial data normalized solution.

Those parameters are explicitly computable from the initial data.

8.1 The initial data normalization

In this section, we define the notion of initial data normalized solution. As we will

show in Theorem 8.3.1, given a seed initial data set and its associated solution S ,

we can find a pure gauge solution G such that the initial data for S ´ G satisfies all

these conditions.

Definition 8.1.1. Consider a seed data set as in Definition 7.1.1 and let S be the re-

sulting solution given by Theorem 7.3.1. We say that S is initial data normalized

if

• the following conditions hold along the null hypersurface C0:

139

1. For the projection to the l “ 0 spherical harmonics:

(1)

κ “ 0 (8.1)


κl“1 “ 0 (8.2)

div pF qβl“1 “ 0 (8.3)

div pF qβl“1

“ 0 (8.4)

3. For the projections to the l ě 1 spherical harmonics:

bA “1

3r3εABBB

´

2σl“1 `pF qρ ˇpF qσl“1

¯

(8.5)

4. For the projection to the l ě 2 spherical harmonics:

pχ “ 0 (8.6)

pχ “ 0 (8.7)

D‹2D‹1pκ, 0q “ 0 (8.8)

• the following conditions hold on the sphere S0:

trγgl“1“ 0 (8.9)

g “ 0 (8.10)

We denote such solutions by S id.

140

We note that the above conditions can all be written explicitly in terms of the

seed data.

We also immediately note by straightforward computation:

Proposition 8.1.0.1. The linearized Kerr-Newman solutions K of Definition 5.2.1

are initial data normalized.

Proof. Condition (8.1) is verified for the linearized Kerr-Newman solutions supported

in l “ 0 spherical harmonics.

Condition (8.5) is verified for the linearized Kerr-Newman solutions supported in

l “ 1 spherical harmonics. Indeed, according to Proposition 5.2.2.1

bA “

ˆ

´8M

r`

4Q2

r2

˙

aεABBBY`“1m “

1

3r3εABBB

´

2σl“1 `pF qρ ˇpF qσl“1

¯

The remaining conditions are trivially verified by any linearized Kerr-Newman solu-

tion K .

8.2 The SU,R-normalization

In this section we define another normalization for a linear perturbation of Reissner-

Nordstrom S . The need for a different normalization than the initial data one will

become clear in Section 9.1, when the decay of the components of a initial data

normalized solution will result incomplete (see Remark 9.1.1).

In order to obtain a complete and optimal decay for all the components of the

solution, we will need to pick gauge conditions ”far away” in the spacetime. In

particular, we construct a gauge solution starting from a sphere SU,R for big U and

big R.

141

We describe here the SU,R normalization, and show in Theorem 8.4.1 that for any

given solution S which is bounded in the past of SU,R we can find a pure gauge

solution GU,R such that S ´ G satisfies all these conditions.

Fix r1 ą rH in Reissner-Nordstrom spacetime with Bondi coordinates pu, r, θ, φq.

Let SU,R be the sphere obtained as the intersection of the hypersurfaces tu “ Uu and

tr “ Ru for some U and R such that R " U , and R " r0. Denote IU,R the null

hypersurface obtained as the ingoing past of SU,R.

I`

C0C0

SU,R

IU,R

tr “ r1u

Figure 8.1: Penrose diagram of Reissner-Nordstrom spacetime with the sphere SU,Rand the null hypersurface IU,R

The characterization of this gauge normalization is related to two new quantities

that we define here.

We define the charge aspect function of a solution S as the scalar function ob-

tained in the following way:

ν “ r4´

div ζ ` 2 pF qρ ˇpF qρ¯

(8.11)

The above definition has a similar structure than the mass aspect function in vacuum

spacetimes, but only depends on the charge of the spacetime (encoded in ˇpF qρ). This

quantity plays a fundamental role in the derivation of the decay for the projection to

the l “ 1 spherical harmonics, which the electromagnetic tensor is responsible for.

142

We also define the mass-charge aspect function of a solution S as the scalar

function obtained in the following way:

µ “ r3´

div ζ ` ρ´ 4 pF qρ ˇpF qρ¯

´ 2r4 pF qρdiv pF qβ (8.12)

Notice that the above definition reduces to the mass aspect function in the absence

of an electromagnetic tensor, i.e. pF qρ “ pF qβ “ 0. In the case of an electrovacuum

spacetime, the function µ depends on both the mass (encoded into ρ) and the charge

(encoded into ˇpF qρ). This generalization plays a fundamental role in the derivation of

decay in Section 10.3.3.

Both quantities, ν and µ verify well-behaved transport equations in the e4 direc-

tion, which is the main reason why they are crucial in the derivation of decay. The

derivation of the equations is obtained in Section A.2 in the Appendix.

We can now define the notion of SU,R-normalization.

Definition 8.2.1. Consider a seed data set as in Definition 7.1.1 and let S be the

resulting solution given by Theorem 7.3.1. Suppose that S is well defined at SU,R for

some U and R. We say that S is SU,R-normalized if

• the following conditions hold along the null hypersurface IU,R:


κl“1 “ 0 (8.13)

κl“1 “ 0 (8.14)

νl“1 “ 0 (8.15)

div bl“1 “ 0 (8.16)

143

2. For the projection to the l ě 2 spherical harmonics:

D‹2D‹1pκ, 0q “ 0 (8.17)

D‹2D‹1pκ, 0q “ 0 (8.18)

D‹2D‹1pµ, 0q “ 0 (8.19)

D‹2b “ 0 (8.20)

where µ is the mass-charge aspect function, as defined in (8.12).

• the following conditions hold on the sphere SU,R:

trγgl“1“ 0 (8.21)

g “ 0 (8.22)

We denote such solutions by S U,R.

We immediately note the following.

Proposition 8.2.0.1. The linearized Kerr-Newman solutions K of Definition 5.2.1

are SU,R-normalized for every U and R.

8.3 Achieving the initial-data normalization for a

general S

In this section, we prove the existence of a pure gauge solution G such that upon

subtracting this to a given S arising from smooth seed data, the resulting solution

is generated by data satisfying all conditions of Definition 8.1.1.

144

Theorem 8.3.1. Consider a seed data set as in Definition 7.1.1 and let S be the

resulting solution given by Theorem 7.3.1. Then there exists a pure gauge solution

G id, explicitly computable from the seed data of S , such that

S id :“ S ´ G id

is initial data normalized. The pure gauge solution G id is unique and arises itself

from the seed data.

Proof. To identify the pure gauge solution G id, it suffices to identify the functions

h, h, a, q1, q2 as in Lemmas 5.1.2.1 and 5.1.3.1. In particular, we will make use of

the conditions in Definition 8.1.1 to determine those functions and their derivative

along the e3 direction, and then we make use of the transport equations required in

Lemmas 5.1.2.1 and 5.1.3.1 to extend those functions in the whole spacetime along

the e4 direction. Using the orthogonal decomposition in spherical harmonics, we can

treat the projection to the l “ 0, l “ 1 and l ě 2 spherical harmonics separately. This

procedure uniquely determines the pure gauge solution G id globally in the spacetime.

Projection to the l “ 0 spherical harmonics - achieving (8.1): Recall that

the only function in the definition of pure gauge solutions which are supported in

l “ 0 spherical harmonics is a. Therefore, we have to globally identify al“0.

We denote(1)

κ0,(1)

κ0 and(1)

Ω0 the functions determined on C0 by the seed initial data

set (recall that(1)

Ω0 is part of the seed initial data set, and(1)

κ0 and(1)

κ0 is uniquely

determined by the seed data according to Theorem 7.3.1).

We define along C0 the following function:

al“0 :“r

2

(1)

κ0

145

Using Lemma 5.1.2.1, we see that the above conditions imply that S1 :“ S ´

G0,0,al“0,0,0 verifies condition (8.1) on C0. The transport equation (5.5) projected to the

l “ 0 spherical harmonics, implies that Bral“0pu, rq “ 0, therefore al“0pu, rq “ al“0puq.

This implies that the above definition of al“0 along C0 determines al“0 globally.

The following choice of pure gauge solutions will be supported in l ě 1, and

therefore will not change condition (8.1).

Projection to the l “ 1 spherical harmonics - achieving (8.2), (8.3) and

(8.4): We identify globally the projection to the l “ 1 spherical harmonics of a, h

and h.

We denote κ0, pF qβ0, pF qβ0

the functions on S0 which are part of the seed initial

data.

We define on S0 the following functions:

hl“1 :“r4

0

2Qdiv pF qβ0l“1

hl“1 :“ ´r4

0

2Qdiv pF qβ

0l“1

al“1 “r0

2

ˆ

κ0l“1 ´2

r20

hl“1 ´

ˆ

1

4κpr0qκpr0q

˙

hl“1 ´1

4κpr0q

2hl“1

˙

Using Lemma 5.1.2.1, we see that the above conditions imply that S2 :“ S1 ´

Ghl“1,hl“1,al“1,0,0 verifies conditions κl“1 “ div pF qβl“1 “ div pF qβl“1“ 0 on S0.

In order to obtain the cancellation along C0, we impose transport equations for

hl“1, hl“1 and al“1. In particular, we have along C0:

∇ 3

ˆ

2Q

r4hl“1

˙

“ ∇ 3div pF qβl“1rS s

∇ 3

ˆ

2Q

r4hl“1

˙

“ ´∇ 3div pF qβl“1rS s

146

The above transport equations together with the above initial conditions uniquely

determine hl“1 and hl“1 along C. We similarly impose the e3 derivative of al“1 to

coincide with the derivative of the right hand side of the above definition.

Transport equation (5.6) then uniquely determines hl“1 globally. The transport

equation (5.5) determines al“1 globally. Then transport equation (5.7) uniquely de-

termines the value of hl“1 globally. This implies that S2 verifies conditions (8.2),

(8.3) and (8.4).

Projection to the l ě 2 spherical harmonics - achieving (8.6), (8.7) and

(8.8): We identify globally the projection to the l ě 2 spherical harmonics of h, h

and a.

We denote D‹2D‹1pκ0, 0q, pχ0, pχ0

the symmetric traceless 2-tensors on S0 which are

determined by the seed initial data.

We define on S0 the following symmetric traceless 2-tensors:

D‹2D‹1ph, 0q :“ ´pχ0

D‹2D‹1ph, 0q :“ ´pχ0

D‹2D‹1pa, 0q :“r0

2

`

D‹2D‹1pκ0, 0q ´ 2D‹2D2pD‹2D‹1ph, 0qq `2

r20

D‹2D‹1ph, 0q

´

ˆ

1

4κpr0qκpr0q

˙

D‹2D‹1ph, 0q ´1

4κpr0q

2D‹2D‹1ph, 0q˘

By Lemma 3.3.4.1, the above conditions uniquely determine the projection to the

l ě 2 spherical modes of h, h and a on S0. Using Lemma 5.1.2.1, we see that the above

conditions imply that S3 :“ S2 ´ Ghlě2,hlě2,alě2,0,0 verifies conditions D‹2D‹1pκ, 0q “

χ “ χ “ 0 on S0.

Applying the e3 derivative to the above definitions we derive transport equa-

tions for D‹2D‹1ph, 0q, D‹2D‹1ph, 0q and D‹2D‹1pa, 0q which uniquely determine them on

147

C0. Then transport equation (5.6) then uniquely determines D‹2D‹1ph, 0q globally and

transport equation (5.5) determines D‹2D‹1pa, 0q globally. Then transport equation

(5.7) uniquely determines the value of D‹2D‹1ph, 0q globally. Again by Lemma 3.3.4.1,

the above conditions uniquely determine the projection to the l ě 2 spherical modes

of h, h and a globally. This implies that S3 verifies in addition conditions (8.6), (8.7)

and (8.8).

Conditions on the metric coefficients - achieving (8.5), (8.9) and (8.10):

With the above, we exhausted the freedom of using Lemma 5.1.2.1, since we globally

determined the functions h, h, a and λ. In particular, the above choices also modified

g, b and trγg on C0, which now differ from the one given by the seed initial data.

In what follows we will achieve the remaining conditions of Definition 8.1.1 using

pure gauge solutions of the form G0,0,0,q1,q2 as in Lemma 5.1.3.1. Observe that these

solutions have all components different from g, b and trγg vanishing, and therefore

do not modify the achieved conditions above.

We identify q1 and q2 on S0.

We define on S0 the following:

pq1ql“1 “ ´1

4r20

trγgl“1rS3s|S0

D‹2D‹1pq1, q2q “1

2r20grS3s|S0

where we denote trγgl“1rS3s|S0 and grS3s|S0 the respective value of trγgl“1

and g of

the solution S3 defined above on the initial sphere S0. By the above discussion, these

values only depend on initial seed data. By Lemma 3.3.4.1, the above conditions

uniquely determine q1 and q2 on S0.

148

Condition (8.5) on C0 determines a transport equation along C0 for q1 and q2:

r2D‹1p∇ 3q1,∇ 3q2qA“ bArS3s ´

1

3r3εABBB

´

2σl“1rS3s `pF qρ ˇpF qσl“1rS3s

¯

which together with (5.8) and (5.9) globally determine q1 and q2. The above conditions

imply that S4 :“ S3´G0,0,0,q1,q2 verifies in addition conditions (8.5), (8.9) and (8.10).

Define G i.d. :“ Gh,h,a,0,0 ` G0,0,0,q1,q2 with h, h, a, q1, q2 determined as above. Then

S i.d. :“ S ´ G i.d.“ S4

verifies all conditions of Definition 8.1.1 and is therefore initial data normalized. By

construction, G i.d. is also uniquely determined.

8.4 Achieving the SU,R normalization for a bounded

S

In this section, we prove the existence of a pure gauge solution G such that upon

subtracting this to a given S , which is assumed to be bounded at the sphere SU,R

for some U and R, the resulting solution satisfies all conditions of Definition 8.2.1.

Observe that we do not need to modify the projection to the l “ 0 spherical harmonics

because such projection is proved to vanish in Section 9.1.1.

Theorem 8.4.1. Consider a seed data set as in Definition 7.1.1 and let S be the

resulting solution given by Theorem 7.3.1. Suppose that the solution S is bounded at

the sphere SU,R.

Then there exists a pure gauge solution G U,R supported in l ě 1 spherical harmon-

149

ics such that

S U,R :“ S ´ G U,R

is SU,R-normalized. The pure gauge solution G U,R is unique and is bounded by the

initial data seed.

Proof. We follow the same pattern as in the proof of Theorem 8.3.1. Since G U,R is

taken to be supported in l ě 1 spherical harmonics it suffices to identify globally the

functions h, h and a with a supported in l ě 1, and q1 and q2. Again, we treat the

projection to the l “ 1 and l ě 2 separately.

Projection to the l “ 1 spherical harmonics - achieving (8.13), (8.14),

(8.15): We identify globally the projection to the l “ 1 spherical harmonics of a, h

and h.

According to Lemma 5.1.2.1, for a pure gauge solution the components κ and κ

verify

κ “ κa` D1D‹1ph, 0q `1

4κκh`

1

4κ2h (8.23)

κ “ ´κa` D1D‹1ph, 0q `ˆ

1

4κ2` ωκ

˙

h`

ˆ

1

4κκ´ ρ

˙

h (8.24)

Multiplying (8.23) by κ and (8.24) by κ and summing them we obtain:

D1D‹1pκh` κh, 0q `ˆ

1

2κκ` ωκ

˙

κh`

ˆ

1

2κκ´ ρ

˙

κh “ κκ` κκ

Setting z “ κh` κh and observing that ωκ “ ´ρ in the background we obtain for a

pure gauge solution

D1D‹1pz, 0q `ˆ

1

2κκ´ ρ

˙

z “ κκ` κκ (8.25)

150

Projecting the above equation to the l “ 1 spherical harmonics, using that D1D‹1 “

´4 and using the Gauss equation (1.32), we obtain an equation for zl“1:

`

´3ρ` 2 pF qρ2˘

zl“1 “ κκrS sl“1 ` κκrS sl“1 (8.26)

The above determines the value of zl“1 along the null hypersurface I U,R.

Multiplying (8.23) by κ and (8.24) by κ and subtracting them we obtain:

2κκa` D1D‹1pκh´ κh, 0q ´ ωκκh` ρκh “ κκ´ κκ

Setting z “ κh´κh and observing that ωκ “ ´ρ we obtain for a pure gauge solution

D1D‹1pz, 0q ` 2κκa “ κκ´ κκ´ ρz (8.27)

According to Lemma 5.1.2.1, for a pure gauge solution the component ν verifies

r´4ν “

ˆ

´1

4κ´ ω

˙

D1D‹1ph, 0q `1

4κD1D‹1ph, 0q ` D1D‹1pa, 0q ` pF qρ2

pκh` κhq

which can be written in terms of z and z as

r´4ν “

ˆ

´1

4κ`

1

4ρr

˙

κ´1D1D‹1pz, 0q ` D1D‹1pa, 0q `1

4ρrκ´1D1D‹1pz, 0q ` pF qρ2z

Multiplying the above by κ and observing that ´14κ ` 1

4rρ “ 1

2r

´

1´ 3Mr`

2Q2

r2

¯

we

obtain

κr´4ν “

ˆ

´1

4κ`

1

4ρr

˙

D1D‹1pz, 0q ` κD1D‹1pa, 0q `1

4ρrD1D‹1pz, 0q ` κ pF qρ2z (8.28)

If R " 3M , then, along IU,R, r " 3M and therefore we can safely multiply (8.27) by

151

12r

´

1´ 3Mr`

2Q2

r2

¯

and subtract (8.28) to obtain:

D1D‹1pa, 0q `ˆ

1

2κκ´ ρ` 4 pF qρ2

˙

a “ r´4ν ´1

2r

ˆ

1´3M

r`

2Q2

r2

˙

κ

`1

2r

ˆ

1´3M

r`

2Q2

r2

˙

κ´1κκ

´

ˆ

1

4rρ

˙

κ´1D1D‹1pz, 0q

´

ˆ

pF qρ2´

1

2r

ˆ

1´3M

r`

2Q2

r2

˙

ρκ´1

˙

z

(8.29)

Projecting to the l “ 1 spherical harmonics we see that the right hand side of (8.29)

is already determined by (8.26). This gives in particular

`

´3ρ` 6 pF qρ2˘

al“1 “ r´4νrS sl“1 ´1

2r

ˆ

1´3M

r`

2Q2

r2

˙

κrS sl“1

`1

2r

ˆ

1´3M

r`

2Q2

r2

˙

κ´1κκrS sl“1

´

ˆ

1

4rρ

˙

κ´12Kzl“1 ´

ˆ

pF qρ2´

1

2r

ˆ

1´3M

r`

2Q2

r2

˙

ρκ´1

˙

zl“1

(8.30)

with known right hand side along I U,R. This determines al“1 along it.

Finally the projection of (8.27) to the l “ 1 spherical harmonics determines zl“1:

2

r2zl“1 “ ´2κκal“1κκrS sl“1 ´ κκrS sl“1 ´ ρzl“1

The value of zl“1 and zl“1 uniquely determine the value of hl“1 and hl“1 along I U,R.

Transport equation (5.6) uniquely determines hl“1 globally, transport equation (5.5)

determines al“1 globally and finally transport equation (5.7) uniquely determines the

value of hl“1 globally. This implies that S ´Ghl“1,hl“1,al“1,0,0 verifies conditions (8.13),

(8.14), (8.15).

Projection to the l ě 2 spherical harmonics - achieving (8.17), (8.18) and

152

(8.19): We identify globally the projection to the l ě 2 spherical harmonics of a, h

and h. We first derive some preliminary relations.

According to Lemma 5.1.2.1, for a pure gauge solution the component µ verifies

r´3µ “

ˆ

´1

4κ´ ω ´ 2r pF qρ2

˙

D1D‹1ph, 0q `1

4κD1D‹1ph, 0q ` D1D‹1pa, 0q

`

ˆ

3

4ρ´

3

2pF qρ2

˙

pκh` κhq

which can be written in terms of z and z as

r´3µ “

ˆ

´1

4κ`

1

4rρ´ r pF qρ2

˙

κ´1D1D‹1pz, 0q `ˆ

1

4rρ´ r pF qρ2

˙

κ´1D1D‹1pz, 0q

`

ˆ

3

4ρ´

3

2pF qρ2

˙

z ` D1D‹1pa, 0q

Multiplying the above by κ and observing that ´14κ` 1

4rρ´ r pF qρ2 “ 1

2r

`

1´ 3Mr

˘

we

obtain

κr´3µ “1

2r

ˆ

1´3M

r

˙

D1D‹1pz, 0q `ˆ

1

4rρ´ r pF qρ2

˙

D1D‹1pz, 0q `ˆ

3

4ρ´

3

2pF qρ2

˙

κz

` κD1D‹1pa, 0q(8.31)

If R " 3M , then, along IU,R, r " 3M and therefore we can safely multiply (8.27) by

12r

`

1´ 3Mr

˘

and subtract (8.31) to obtain:

D1D‹1pa, 0q `ˆ

1


˙

a “ r´3µ´1

2r

ˆ

1´3M

r

˙

κ`1

2r

ˆ

1´3M

r

˙

κ´1κκ

´

ˆ

1

4rρ´ r pF qρ2

˙

κ´1D1D‹1pz, 0q

´

ˆ

3

4ρ´

3

2pF qρ2

´1

2r

ˆ

1´3M

r

˙

ρκ´1

˙

z

(8.32)

To verify conditions (8.17), (8.18) and (8.19) we are interested in determining

153

D‹2D‹1pz, 0q, D‹2D‹1pz, 0q and D‹2D‹1pa, 0q on IU,R.

We apply the operator D‹2D‹1 to (8.25), which translates in the following relation

along IU,R:

D‹2D‹1D1D‹1pz, 0q `ˆ

1

2κκ´ ρ

˙

D‹2D‹1pz, 0q “ κD‹2D‹1pκrS s, 0q ` κD‹2D‹1pκrS s, 0q

By (1.8) we have that

D‹2D‹1D1 “ p2D‹2D2 ` 2KqD‹2 (8.33)

which therefore implies the following relation for D‹2D‹1pz, 0q:

ˆ

2D‹2D2 ` 2K `1

2κκ´ ρ

˙

D‹2D‹1pz, 0q “ κD‹2D‹1pκrS s, 0q ` κD‹2D‹1pκrS s, 0q

By Gauss equation, we have

2D‹2D2 ` 2K `1

2κκ´ ρ “ 2D‹2D2 ` 2

ˆ

´1

4κκ´ ρ` pF qρ2

˙

`1

2κκ´ ρ

“ 2D‹2D2 ´ 3ρ` 2 pF qρ2

Define E to be the operator E :“ 2D‹2D2´3ρ`2 pF qρ2 on symmetric traceless 2-tensors.

Then the above relation gives

EpD‹2D‹1pz, 0qq “ κD‹2D‹1pκrS s, 0q ` κD‹2D‹1pκrS s, 0q (8.34)

We show that the operator E is coercive.

Lemma 8.4.0.1. Let E be the operator defined as E :“ 2D‹2D2´3ρ`2 pF qρ2. For any

154

symmetric traceless two tensor θ we have

ż

S

θ ¨ Eθ ě 4

r2

ż

S

|θ|2

Proof. We compute

ż

S

θ ¨ Eθ “ż

S

θ ¨ p2D‹2D2 ´ 3ρ` 2 pF qρ2qθ “

ż

S

2|D2θ|2 ` p´3ρ` 2 pF qρ2q|θ|2

Using the standard Poincare inequality on spheres andş

S|∇ θ|2`2K|θ|2 “ 2

ş

S|D2θ|2,

we have thatş

S|D2θ|2 ě

ş

S2K|θ|2, and therefore

ż

S

θ ¨ Eθ ěż

S

ˆ

4

r2`

6M

r3´

4Q2

r4

˙

|θ|2

Observe that 6Mr3´

4Q2

r4ě 2M2

r3for all r ą M and |Q| ă M . We therefore obtain the

inequality.

The above Lemma shows that (8.34) uniquely determines D‹2D‹1pz, 0q along IU,R.

Applying the operator D‹2D‹1 to (8.32) we obtain on IU,R

D‹2D‹1D1D‹1pa, 0q `ˆ

1


˙

D‹2D‹1pa, 0q “ RHSpµrS s, κrS s, κrS s, D‹2D‹1pz, 0qq

where the right hand side depends on the argument, which are determined along

IU,R. Using (8.33), we obtain

`

2D‹2D2 ´ 3ρ` 6 pF qρ2˘

D‹2D‹1pa, 0q “ RHSpµrS s, κrS s, κrS s, D‹2D‹1pz, 0qq(8.35)

The above operator is a slight modification of E (which is even more positive) and

possesses an identical Poincare inequality as in Lemma 8.4.0.1. The above relation

155

therefore implies that D‹2D‹1pa, 0q is uniquely determined along IU,R by the above

imposition.

Finally, applying the operator D‹2D‹1 to (8.27) and using (8.33) we obtain on IU,R:

p2D‹2D2 ` 2KqD‹2D‹1pz, 0q “ κD‹2D‹1pκrS s, 0q ´ κD‹2D‹1pκrS s, 0q

´ ρz ´ 2κκD‹2D‹1pa, 0q(8.36)

where the right hand side has already been determined above. The above operator is

clearly coercive. Indeed, using elliptic estimate (1.7) we have

ż

S

θ ¨ p2D‹2D2 ` 2Kqθ “

ż

S

2|D2θ|2 ` 2K|θ|2 ě4

r2

ż

S

|θ|2

The above relation therefore implies that D‹2D‹1pz, 0q is uniquely determined along

IU,R.

To check that these tensors are smooth along IU,R, we show that their e3 derivative

is smooth along the null hypersurface. For instance, suppose θ is a symmetric traceless

2-tensor which verifies

Epθq “ F

where F is a smooth known function on IU,R. We compute

rE ,∇ 3s “ p2D‹2D2 ´ 3ρ` 2 pF qρ2qp∇ 3θq ´∇ 3pp2D‹2D2 ´ 3ρ` 2 pF qρ2

qθq

“ κp2D‹2D2 ´9

2ρ` pF qρ2

qθ

“ κF ´ κˆ

3

2ρ` pF qρ2

˙

θ

156

Therefore we obtain

Ep∇ 3θq ` κ

ˆ

3

2ρ` pF qρ2

˙

θ “ ∇ 3F ` κF

Applying the operator E to the above we obtain

E2p∇ 3θq “ Ep∇ 3F ` κFq ´ κ

ˆ

3

2ρ` pF qρ2

˙

F

This shows that ∇ 3θ is smoothalong IU,R. A similar computation applied for the

modified version of E applied to D‹2D‹1pa, 0q and the operator applied to D‹2D‹1pz, 0q.

The above choices uniquely determine the projection to the l ě 2 spherical har-

monics of h, h and a on IU,R and as above, integrating in order the transport equation

for h, a and then for h, we show that they are globally uniquely determined.

By construction,

S1 “ S ´ Gh,h,a,0,0

verifies conditions (8.13), (8.14), (8.15), (8.17), (8.18) and (8.19).

Conditions on the metric coefficients - achieving (8.16), (8.20), (8.21) and

(8.22): With the above, we exhausted the freedom of using Lemma 5.1.2.1, since

we globally determined the functions h, h, a. In particular, the above choices also

modified g, b and trγg on IU,R. In what follows we will achieve the remaining condi-

tions using pure gauge solutions of the form G0,0,0,q1,q2 as in Lemma 5.1.3.1. Observe

that these solutions have all components different from g, b and trγg vanishing, and

therefore do not modify the achieved conditions above.

The conditions here imposed are almost identical to the one imposed to the met-

ric coefficient in the initial data normalization. The procedure to find q1 and q2 is

157

identical. We sketch it here.

We define on SU,R the following:

pq1ql“1 “ ´1

4r20

trγgl“1rS1s|SU,R

D‹2D‹1pq1, q2q “1

2r20grS1s|SU,R

By Lemma 3.3.4.1, the above conditions uniquely determine q1 and q2 on SU,R.

Conditions (8.16) and (8.20) on IU,R determine a transport equation along IU,R

for q1 and q2:

r2D‹2D‹1p∇ 3q1,∇ 3q2q “ D‹2brS1s

2p∇ 3q1ql“1 “ div brS1sl“1

which together with (5.8) and (5.9) globally determine q1 and q2.

Define G U,R :“ Gh,h,a,q1,q2 with h, h, a, q1, q2 determined as above. Then

S U,R :“ S ´ G U,R

verifies all conditions of Definition 8.2.1 and is therefore SU,R-normalized. By con-

struction, G U,R supported in l ě 1 is uniquely determined.

8.5 The Kerr-Newman parameters in l “ 0, 1 modes

The initial data normalization will allow to identify the Kerr-Newman parameters

from the initial data seed. Notice that, in contrast with the linear stability of Schwarz-

schild in [16], the projection of the initial data normalized solution to the l “ 0, 1

158

modes is not exhausted by the linearized Kerr-Newman solution. Because of the pres-

ence of the electromagnetic radiation, there is decay of the components at the level

of the l “ 1 mode (see Section 10.3.1).

We define the Kerr-Newman parameters, which are read off at the initial sphere

S0 of radius r0.

Definition 8.5.1. Let S and S id as in Theorem 8.3.1. We denote K id the linearized

Kerr-Newman solution KpM,Q,b,aq where the parameters M,Q, b, ai are given by

Q “ r20

(1)

pF qρ|S0 , M “ ´r3

0

2

(1)

ρ|S0 ` 2Qr0

(1)

pF qρ|S0 , b “ r20

(1)

pF qσ|S0 ,

a´1 “ˇpF qσl“1,m“´1|S0 , a0 “

ˇpF qσl“1,m“0|S0 , a1 “ˇpF qσl“1,m“1|S0

where the above quantities refer to the initial data normalized solution S id.

Observe that the Kerr-Newman parameters in this definition are explicitably com-

putable from the seed initial data.

159

Chapter 9

Proof of boundedness

In this chapter, we prove boundedness of a linear perturbation of Reissner-Nordstrom

spacetime S which is initial data-normalized, upon subtracting a member of the

linearized Kerr family.

In Section 9.1, we use the initial data normalization to prove boundedness. In the

process of obtaining boundedness, we obtain decay for some components, and non

optimal decay for other components. We make use of this boundedness statement

to use the SU,R-normalization, and in Section 9.2 we prove that the gauge solution

decays and is controlled by initial data.

9.1 Initial data normalization and boundedness

Here we state the result proved in this chapter.

Let S be a linear gravitational and electromagnetic perturbation around Reissner-

Nordstrom spacetime pM, gM,Qq, with |Q| ! M , arising from regular asymptotically

flat initial data. Let S id be the initial data normalized solution associated to S by

Theorem 8.3.1 and let K id the linearized Kerr-Newman solution associated to S id

160

as in Definition 8.5.1. Define

S id,K :“ S id´K id

“ S ´ G id´K id (9.1)

We then prove the following.

Proposition 9.1.0.1. The solution S id,K has all bounded components. Moreover,

its projection to the l “ 0 spherical harmonics vanishes.

The proof of the Proposition makes use of the gauge conditions imposed in the

initial data normalization to then integrate forward the transport equations. The

process of integrating forward from a bounded r necessarily implies that some com-

ponents (namely ξ and ω) would not decay in r. For this reason, with this procedure

we fail to obtain decay in r for ξ and ω. In the next chapter we introduce the SU,R-

normalization, through which we can integrate backward from an unbounded r. This

allows to obtain the optimal decay in r and u (i.e. as given in Theorem 10.1.1).

In Section 9.1.1 we prove that the projection to the l “ 0 spherical mode of such

a solution vanishes.

In Section 9.1.2 and in Section 9.1.3 we prove boundedness and decay for the

projection to the l “ 1 mode and l ě 2 modes respectively.

Finally in Section 9.1.4 we derive decay for the quantities involved in the e3 di-

rections, i.e. ξ, η and ω for which the decay is not optimal. This lack of optimality

is the reason to use the SU,R-normalization later.

We first summarize the main properties of the solution S id,K .

1. Since by Proposition 8.1.0.1, the linearized Kerr-Newman solution is initial data

normalized, then S id,K is initial data normalized, i.e. conditions (8.1)-(8.10)

are verified.

161

2. By Definition 8.5.1 of the linearized Kerr-Newman solution K id, the solution

S id,K verifies in addition (recall condition (8.5)):

(1)

pF qρ “(1)

ρ “(1)

pF qσ “ ˇpF qσl“1 “ 0 on S0 (9.2)

pdiv bql“1 “ D‹2b “ 0, on C0 (9.3)

pcurl bql“1 “ ´2

3r34 σl“1 “

4

3rσl“1 on C0 (9.4)

We show that the above conditions imply the vanishing of the projection to the

l “ 0 spherical mode, and the boundedness of the projection to the l ě 1 spherical

mode.

In the following, we need to integrate transport equations from the initial data

hypersurface C0 in the e4 directions forward. We summarize the procedure in the

following lemma.

We denote A À B if there exists an universal constant C depending on the initial

data such that A ď CB.

Lemma 9.1.0.1. If f verifies the transport equation

∇ 4f `p

2κf “ F

and f and F satisfy the following estimates:

|f | À u´1`δ on C0 (9.5)

|F | À mintr´q´1u´12`δ, r´qu´1`δu on tr ą rHu (9.6)

162

for some q ě 0, then for any u ě u0 and r ą rH,

|f | À mintr´mintp,quu´12`δ, r´mintp,q´1uu´1`δu

Proof. According to Proposition 2.3.0.1, the transport equation verified by f is equiv-

alent to

∇ 4prpfq “ rpF

Using (3.16), the transport equation becomes

Brprpfq “ rpF

Consider now a fixed u ě u0. The null hypersurface of fixed u intersects C0 at a

certain r “ rpuq in the sphere Su,rpuq. We now integrate the above equation along the

fixed u hypersurface from the sphere Su,rpuq on C0 to the sphere Su,r for any r ě rpuq.

We obtain

rpfpu, rq “ rpuqpfpu, rpuqq `

ż r

r

λpF pu, λqdλ

If conditions (9.5) and (9.6) are satisfied, then |fpu, rpuqq| À u´1`δ and |F pu, λq| À

mintλ´q´1u´12`δ, λ´qu´1`δu on tr ě ru, which gives

rp|fpu, rq| À rpuqpu´1`δ`

ż r

r

mintλp´q´1u´12`δ, λp´qu´1`δudλ

À rpuqpu´1`δ`mintrp´qu´12`δ, rp´q`1u´1`δ

u `mintrp´qu´12`δ, rp´q`1u´1`δu

Since by construction rpuq ď r0 for every u ě u0, we can bound the right hand side

163

by :

rp|fpu, rq| À u´1`δ`mintrp´qu´12`δ, rp´q`1u´1`δ

u

where the constant is understood to depend on the sphere of the initial sphere r0.

Diving by rp, we prove the lemma.

9.1.1 The projection to the l “ 0 mode

We prove here that the projection to the l “ 0 spherical mode of the solution S id,K

defined in (9.1) vanishes.

Vanishing of the l “ 0 mode on S0

1. From (8.1) and (9.2), we have on S0:

(1)

κ “(1)

pF qρ “(1)

ρ “(1)

pF qσ “ 0

2. Applying Gauss equation (4.37) to the sphere S0 we obtain(1)

κ “ 0, and therefore

from (4.12) we obtain(1)

Ω “ 0.

3. Condition (8.1) holds on C0, therefore it implies ∇ 3(1)

κ “ 0 on S0. Restricting

(4.27) to S0 we obtain that(1)

ω “ 0 on S0.

Vanishing of the l “ 0 mode on C0

1. From (8.1) we have on C0:

(1)

κ “ 0

164

2. From the transport equations (4.44) and (4.46) and the vanishing initial data

on S0 for(1)

pF qσ and(1)

pF qρ, we obtain(1)

pF qσ “(1)

pF qρ “ 0 on C0.

3. From the transport equation (4.58) and the vanishing initial data on S0 for(1)

ρ,

we obtain(1)

ρ “ 0 on C0.

4. From Gauss equation (4.37) and (8.1) we obtain(1)

κ “ 0 on C0, and therefore

from (4.12) we obtain(1)

Ω “ 0 on C0.

5. Restricting (4.27) to C0 we obtain that(1)

ω “ 0 on C0.

Vanishing of the l “ 0 mode everywhere

1. From (8.1) and (4.28) we have globally:

(1)

κ “ 0

2. From the transport equations (4.45) and (4.47) and the vanishing initial data

on C0 for(1)

pF qσ and(1)

pF qρ, we obtain(1)

pF qσ “(1)

pF qρ “ 0 globally.

3. From the transport equation (4.59) and the vanishing initial data on C0 for(1)

ρ,

we obtain(1)

ρ “ 0 globally.

4. From Gauss equation (4.37) we obtain(1)

κ “ 0 globally.

5. From (4.31) and the vanishing initial data on C0 we obtain that(1)

ω “ 0 globally.

6. From (4.12) and the vanishing initial data on C0 we obtain that(1)

Ω “ 0 globally.

The projection to the l “ 0 spherical mode of an initial data normalized solution is

therefore exhausted by a linearized Reissner-Nordstrom solution, with no non-trivial

decay supported in this spherical mode.

165


In contrast with the case of linear stability of Schwarzschild in [16], in the linear

stability of Reissner-Nordstrom, because of the presence of the electromagnetic radi-

ation, we expect the projection to the l “ 1 mode of the solution not to be exhausted

by a pure gauge and a linearized Kerr-Newman solution. We indeed show that there

is also decay at the level of the projection to the l “ 1 mode.

To obtain decay for the projection to the l “ 1 spherical mode, we make use

of Theorem 6.5.1 stating the decay for the gauge-invariant quantities β, β, p. In

particular, we show that we can express all the remaining quantities in terms of

only pF qβ, pF qβ and κ and the gauge-invariant quantities already estimated. This will

simplify the computations and the derivation of the estimates for all quantities.

Notation We denote that a quantity ξ is Opr´p´1u´12`δ, r´pu´1`δq if

|ξ| À mintr´p´1u´12`δ, r´pu´1`δu for all u ě u0 and r ą rH

In particular, we write ξ1 “ ξ2 `Opr´p´1u´12`δ, r´pu´1`δq if

ξ1 “ ξ2 ` ξ3

with ξ3 “ Opr´p´1u´12`δ, r´pu´1`δq.

Since the following relations will be used later in the proof of the optimal decay,

we summarize them in the following Proposition. To derive those, we only use elliptic

relations, and not transport equations which will be exploited later.

166

Proposition 9.1.2.1. The following relations hold true for all u ě u0 and r ą rH:

pdiv βql“1 “3ρ

2 pF qρpdiv pF qβql“1 Òpr

´4´δu´12`δ, r´3´δu´1`δq (9.7)


2 pF qρpdiv pF qβql“1 Òpr

´3u´1`δq (9.8)

pdiv ζql“1 “1

rκl“1 ` r

ˆ

3ρ

2 pF qρ´

pF qρ

˙

pdiv pF qβql“1 Òpr´3´δu´12`δ, r´2´δu´1`δ

q(9.9)

κl“1 “ ´1

2rκκl“1 ´

1

2r3κ

ˆ

3ρ

2 pF qρ´

pF qρ

˙

pdiv pF qβql“1 ` r2

ˆ

3ρ

2 pF qρ´

pF qρ

˙

pdiv pF qβql“1

Òpr´1u´1`δq

(9.10)

ρl“1 “1

4r2κ

ˆ

3ρ

2 pF qρ`

pF qρ

˙

pdiv pF qβql“1 ´1

2r

ˆ

3ρ

2 pF qρ`

pF qρ

˙

pdiv pF qβql“1

Òpr´2u´1`δq

(9.11)

ˇpF qρl“1 “1

4r2κpdiv pF qβql“1 ´

1

2rpdiv pF qβql“1 Òpr

´1u´1`δq (9.12)

Proof. Recall the definition of the gauge-invariant quantities β and β as defined in

(5.13). By taking the divergence and projecting to the l “ 1 spherical mode, we have

pdiv βql“1 “ 2 pF qρpdiv βql“1 ´ 3ρpdiv pF qβql“1

pdiv βql“1 “ 2 pF qρpdiv βql“1 ´ 3ρpdiv pF qβql“1

167

We can in particular isolate β and β and obtain:


2 pF qρpdiv pF qβql“1 `

1

2 pF qρpdiv βql“1


2 pF qρpdiv pF qβql“1 `

1

2 pF qρpdiv βql“1

Using the estimates for β and β as in (6.14) and (6.17), we obtain (9.7) and (9.8).

Applying div to Codazzi equation (4.26) and projecting to the l “ 1 spherical

harmonics, since div div pχl“1 “ 0 and κ “ 2r

we obtain

pdiv ζql“1 “1

2rD1D‹1pκ, 0ql“1 ` rpdiv βql“1 ´ r

pF qρpdiv pF qβql“1

Using (1.8) to project the laplacian to the l “ 1, and using (9.7), we obtain (9.9).

Applying div to Codazzi equation (4.25) and projecting to the l “ 1 spherical

harmonics, since div div pχl“1“ 0 we obtain

1

2D1D‹1pκ, 0ql“1 “ ´

1

2κpdiv ζql“1 ` pdiv βql“1 ´

pF qρpdiv pF qβql“1

Using (1.8) to project the laplacian to the l “ 1, and using (9.8), we obtain (9.10).

Projecting Gauss equation (4.41) to the l “ 1 spherical harmonics and using (4.6),

we have

ρl“1 “ ´1

4κκl“1 ´

1

4κκl“1 ` 2 pF qρ ˇpF qρl“1

“1

4r2κ

ˆ

3ρ

2 pF qρ´

pF qρ

˙


2r

ˆ

3ρ

2 pF qρ´

pF qρ

˙

pdiv pF qβql“1

`2 pF qρ ˇpF qρl“1 `Opr´2u´1`δ

q

Commuting the expression for p given by (A.2), with div and projecting to the

168

l “ 1 spherical harmonics we obtain

pdiv pql“1

r5“ 2 pF qρD1D‹1p´ρ, σql“1 ` p3ρ´ 2 pF qρ2

qD1D‹1p ˇpF qρ, ˇpF qσql“1

`2 pF qρ2pκpdiv pF qβql“1 ´ κpdiv pF qβql“1q

“ 2 pF qρ

ˆ

´2

r2ρl“1

˙

` p3ρ´ 2 pF qρ2q

ˆ

2

r2

ˇpF qρl“1

˙

`2 pF qρ2pκpdiv pF qβql“1 ´ κpdiv pF qβql“1q

Making use of the estimate (6.11) for p and of the above relation for ρl“1 we have

p3ρ´ 2 pF qρ2qp2 ˇpF qρl“1q “ 4 pF qρpρl“1q ´ 2r2 pF qρ2

pκpdiv pF qβql“1 ´ κpdiv pF qβql“1q

Òpr´4u´1`δq

“1

2r2κ

`

3ρ´ 2 pF qρ2˘

pdiv pF qβql“1 ´ r`

3ρ´ 2 pF qρ2˘

pdiv pF qβql“1

`8 pF qρ2 ˇpF qρl“1 ´ 2r2 pF qρ2pκpdiv pF qβql“1 ´ κpdiv pF qβql“1q

Òpr´4u´1`δq

which gives

p3ρ´ 6 pF qρ2qp2 ˇpF qρl“1q “

1

2r2κ

`

3ρ´ 6 pF qρ2˘

pdiv pF qβql“1 ´ r`

3ρ´ 6 pF qρ2˘

pdiv pF qβql“1

Òpr´4u´1`δq

Putting together the above, we finally obtain (9.11) and (9.12).

We now obtain control over κl“1, pdiv pF qβql“1, pdiv pF qβql“1 by using transport

equations.

1. Recall Lemma 3.3.4.4 for the commutation of the ∇ 4 derivative with the pro-

169

jection to the l “ 1 spherical harmonics. From (4.32) we obtain

∇ 4pκl“1q “ p∇ 4κql“1 “ ´κκl“1

Together with condition (8.2), this implies

κl“1 “ 0 for all u ě u0 and r ą rH (9.13)

2. From equation (A.3) commuted with div and the estimates (6.14) for β we have

∇ 4div pF qβ ` 2κdiv pF qβ “ Opr´4´δu´12`δ, r´3´δu´1`δq

Using condition (8.3), we can apply Lemma 9.1.0.1 to the above equation pro-

jected to the l “ 1 spherical harmonics for p “ 4 and q “ 3` δ, we obtain

|pdiv pF qβql“1| À mintr´3´δu´12`δ, r´2´δu´1`δu for all u ě u0 and r ą rH(9.14)

3. Using (9.9) we deduce

|pdiv ζql“1| À mintr´3´δu´12`δ, r´2´δu´1`δu for all u ě u0 and r ą rH(9.15)

4. From commuting (4.43) with div we obtain

∇ 4div pF qβ ` κdiv pF qβ “ D1D‹1p ˇpF qρ,´ ˇpF qσq ` 2 pF qρdiv ζ

170

Projecting the above to the l “ 1 spherical harmonics we have

∇ 4ppdiv pF qβql“1q ` κpdiv pF qβql“1 “2

r2

ˇpF qρl“1 ` 2 pF qρpdiv ζql“1

Using (9.15) and (9.12) we simplify it to

∇ 4ppdiv pF qβql“1q ` κpdiv pF qβql“1 “2

r2p´

1

2rpdiv pF qβql“1q `Opr

´3u´1`δq

which gives

∇ 4ppdiv pF qβql“1q `3

2κpdiv pF qβql“1 “ Opr´3u´1`δ

q

Integrating the above equation using (8.4) we obtain

|pdiv pF qβql“1| À r´2u´1`δ for all u ě u0 and r ą rH (9.16)

5. The decay obtained for κl“1, pdiv pF qβql“1, pdiv pF qβql“1 allows to deduce the

following decays for all u ě u0 and r ą rH using Proposition 9.1.2.1:

|pdiv βql“1| À mintr´4´δu´12`δ, r´3´δu´1`δu

|pdiv βql“1| À r´3u´1`δ

|κl“1| À r´1u´1`δ

|ρl“1| À r´2u´1`δ

|ˇpF qρl“1| À r´1u´1`δ

171

The curl part

We now obtain decay for the curl part of the projection to the l “ 1 spherical

harmonics. Observe that is in this part that the linearized Kerr-Newman solutions

live. We derive general elliptic relations for the curl part in the following proposition,

where we express all the relevant quantities to the pcurl pF qβql“1.


pcurl βql“1 “3ρ

2 pF qρpcurl pF qβql“1 Òpr

´4´δu´12`δ, r´3´δu´1`δq (9.17)

pcurl ζql“1 “ r

ˆ

3ρ

2 pF qρ´

pF qρ

˙

pcurl pF qβql“1 Òpr´3´δu´12`δ, r´2´δu´1`δ

q(9.18)

pcurl pF qβql“1 “1

2κrpcurl pF qβql“1 Òpr

´2u´1`δq (9.19)

pcurl βql“1 “3ρ

4 pF qρκrpcurl pF qβql“1 Òpr

´3u´1`δq (9.20)

pcurl ηql“1 “ r

ˆ

3ρ

2 pF qρ´

pF qρ

˙


q(9.21)

σl“1 “ r

ˆ

3ρ

2 pF qρ´

pF qρ

˙


q(9.22)

ˇpF qσl“1 “ ´rpcurl pF qβql“1 Òpr´1u´1`δ

q (9.23)

Proof. By taking the curl of the definition of the gauge-invariant quantity β and using

their estimates we obtain (9.17).

Applying curl to Codazzi equation (4.26) and projecting to the l “ 1 spherical

harmonics, since curl div pχl“1 “ 0 and curl D‹1pκ, 0q “ 0 we obtain

pcurl ζql“1 “ rpcurl βql“1 ´ rpF qρpcurl pF qβql“1

Using (9.17), we obtain (9.18).

Applying curl to Codazzi equation (4.25) and projecting to the l “ 1 spherical

172

harmonics, we obtain

0 “ ´1

2κpcurl ζql“1 ` pcurl βql“1 ´

pF qρpcurl pF qβql“1

Using (9.20) and (9.18), we obtain

0 “ ´1

2κr

ˆ

3ρ

2 pF qρ´

pF qρ

˙

pcurl pF qβql“1 `3ρ

2 pF qρpcurl pF qβql“1 ´

pF qρpcurl pF qβql“1

`Opr´3u´1`δq

which gives (9.19).

By taking the curl of the definition of the gauge-invariant quantity β and using

(9.19) we obtain (9.20).

Projecting (4.39) and (4.40) to the l “ 1 spherical harmonics, we obtain (9.21)

and (9.22).

Commuting the expression for p given by (A.2), with curl and projecting to the

l “ 1 spherical harmonics we obtain

pcurl pql“1

r5“ 2 pF qρcurl D‹1p´ρ, σql“1 ` p3ρ´ 2 pF qρ2

qcurl D‹1p ˇpF qρ, ˇpF qσql“1

`2 pF qρ2pκpcurl pF qβql“1 ´ κpcurl pF qβql“1q

Using that curl D‹1p´ρ, σql“1 “ ´4 σl“1 “2r2σl“1, and making use of the estimate

(6.11) for p and of the above relation for σl“1 and curl pF qβl“1

we have

2 pF qρpr

ˆ

3ρ

2 pF qρ´

pF qρ

˙

pcurl pF qβql“1q ` p3ρ´ 2 pF qρ2q

ˇpF qσl“1 “ Opr´4u´1`δq

which gives (9.23).

173

We now obtain control over pcurl pF qβql“1 by using transport equations.

1. Recall that we have by (9.2) that ˇpF qσl“1 “ 0 on S0. We now consider the

projection to the l “ 1 spherical harmonics of the equation (4.50). According

to Lemma 3.3.4.4, we obtain

∇ 3pˇpF qσl“1q ` κ

ˇpF qσl“1 “ pcurl pF qβql“1

Using (9.19) and (9.23) we obtain

∇ 3pˇpF qσl“1q ` κ

ˇpF qσl“1 “1

2κrcurl pF qβl“1 `A “ ´

1

2κ ˇpF qσl“1 `A

where A is a gauge-invariant quantity that has the pointwise estimateOpr´2u´1`δq

as indicated in Proposition 9.1.2.2. The above equation reduces then to

∇ 3pr3 ˇpF qσl“1q “ r3A

According to the estimates obtained in Theorem 6.5.1, the gauge invariant set of

quantities A also have a consistent L2 estimates on spacelike hypersurfaces and

along null hypersurfaces as in (6.7). In particular, we haveş

C0|A |2 ď u´2`2δ.

Integrating then the above equation over C0 from S0 and using the vanishing of

ˇpF qσ on S0, we can bound the right hand side by its L2 norm and obtain decay

in u along C0 for ˇpF qσl“1, i.e.

|ˇpF qσl“1| À u´1`δalong C0 (9.24)

2. By (9.23) restricted to C0 we obtain |curl pF qβl“1| À u´1`δ along C0. From

174

equation (A.3) commuted with curl and the estimates (6.14) for β we have

∇ 4curl pF qβ ` 2κcurl pF qβ “ Opr´4´δu´12`δ, r´3´δu´1`δq

Projecting the equation to the l “ 1 spherical harmonics we obtain

∇ 4ppcurl pF qβql“1q ` 2κpcurl pF qβql“1 “ Opr´4´δu´12`δ, r´3´δu´1`δq

Integrating the above equation using Lemma 9.1.0.1 with p “ 4 and q “ 3` δ,

we obtain

|pcurl pF qβql“1| À mintr´3´δu´12`δ, r´2´δu´1`δu for all u ě u0 and r ą rH(9.25)

3. Using (9.25) and Proposition 9.1.2.2 we obtain for all u ě u0 and r ą rH:

|pcurl βql“1| À mintr´4´δu´12`δ, r´3´δu´1`δu (9.26)

|pcurl ζql“1| À mintr´3´δu´12`δ, r´2´δu´1`δu (9.27)

|pcurl pF qβql“1| À r´2u´1`δ (9.28)

|pcurl βql“1| À r´3u´1`δ (9.29)

|pcurl ηql“1| À r´2u´1`δ (9.30)

|σl“1| À r´2u´1`δ (9.31)

|ˇpF qσl“1| À r´1u´1`δ (9.32)

4. Using (9.25) and (9.26) and apply Lemma 9.1.0.1 to (4.51) with p “ 2 and

175

q “ 2` δ and to (4.63) with p “ 3 and q “ 3` δ, we obtain

|σl“1| À r´3u´12`δ (9.33)

|ˇpF qσl“1| À r´2u´12`δ (9.34)

9.1.3 The projection to the l ě 2 modes

To obtain decay for the projection to the l ě 2 spherical modes, we make use of the

decay obtained in Theorem 6.5.1 for the gauge-invariant quantities f, f, β, β, qF, p.

In particular, we can express all the remaining quantities in terms of only pχ, pχ and

D‹2D‹1pκ, 0q and the gauge-invariant quantities already estimated.

As for the l “ 1 projection, we summarize those relations in the following Propo-

sition. We only use elliptic relations, and not transport equations which will be

exploited later.


D‹2 pF qβ “ ´pF qρpχÒpr´3´δu´12`δ, r´2´δu´1`δ

q (9.35)

D‹2 pF qβ “pF qρpχÒpr´2u´1`δ

q (9.36)

D‹2β “ ´3

2ρpχÒpr´4´δu´12`δ, r´3´δu´1`δ

q (9.37)

D‹2β “3

2ρpχÒpr´3u´1`δ

q (9.38)

D‹2ζ “ r

ˆ

D‹2D2pχ`ˆ

´3

2ρ` pF qρ2

˙

pχ

˙

`1

2rD‹2D‹1pκ, 0q

Òpr´3´δu´12`δ, r´2´δu´1`δq

(9.39)

176

D‹2D‹1p ˇpF qρ, ˇpF qσq “ ´1

2pF qρ

`

κpχ` κpχ˘

Òpr´4u´12`δ, r´3u´1`δq (9.40)

D‹2D‹1p´ρ, σq “

ˆ

3

4ρ`

1

2pF qρ2

˙

`

κpχ` κpχ˘

Òpr´4u´1`δq (9.41)

D‹2D‹1pκ, 0q “ ´2

ˆ

D‹2D2pχ`ˆ

´3

2ρ` pF qρ2

˙

pχ

˙

´ rκ

ˆ

D‹2D2pχ`ˆ

´3

2ρ` pF qρ2

˙

pχ

˙

´1

2rκD‹2D‹1pκ, 0q Òpr´3u´1`δ

q

(9.42)

Proof. Recall the definition of f and f as defined in (5.10). We can then express pF qβ

and pF qβ in terms of pχ and pχ respectively:

D‹2 pF qβ “ ´pF qρpχ` f

D‹2 pF qβ “pF qρpχ` f

Using the estimates for f and f as in (6.13) and (6.16), we obtain (9.35) and (9.36).

Using the definition of β and β and the above relations to express β and β in

terms of pχ and pχ, we obtain

2 pF qρD‹2β “ D‹2β ` 3ρD‹2 pF qβ “ D‹2β ` 3ρp´ pF qρpχ` fq

2 pF qρD‹2β “ D‹2β ` 3ρD‹2 pF qβ “ D‹2β ` 3ρp pF qρpχ` fq

which gives

D‹2β “ ´3

2ρpχ` pF qρ´1

p1

2D‹2β `

3

2ρfq

D‹2β “3

2ρpχ` pF qρ´1

p1

2D‹2β `

3

2ρfq

Using the estimates for β, β as in (6.16), (6.14), (6.17), we obtain (9.37) and (9.38).

177

Using the Codazzi equation (4.26), we express ζ in terms of κ and pχ. Indeed we

obtain

ζ “ rD2pχ`1

2rD‹1pκ, 0q ` rβ ´ r pF qρ pF qβ

Applying the operator D‹2 to the above and using (9.35) and (9.37) we obtain

D‹2ζ “ rD‹2D2pχ`1

2rD‹2D‹1pκ, 0q ` rD‹2β ´ r pF qρD‹2 pF qβ

“ rD‹2D2pχ`1

2rD‹2D‹1pκ, 0q ` rp´

3

2ρpχ`mintr´4´δu´12`δ, r´3´δu´1`δ

uq

´r pF qρp´ pF qρpχ`mintr´3´δu´12`δ, r´2´δu´1`δuq

which gives (9.39).

We use the alternative expression for qF given by (A.1) to express ˇpF qρ and ˇpF qσ

in terms of pχ and pχ:

D‹2D‹1p ˇpF qρ, ˇpF qσq “ ´1

2pF qρ

`

κpχ` κpχ˘

´qF

r3

Using the estimate for qF given by (6.10), we obtain (9.40).

We use the alternative expression for p given by (A.2), and the above relations,

to express ρ and σ in terms of pχ and pχ. Using the estimate for p given by (6.11) we

have

2 pF qρD‹1p´ρ, σq “ p´3ρ` 2 pF qρ2qD‹1p ˇpF qρ, ˇpF qσq ` 2 pF qρ2

pκ pF qβ ´ κ pF qβq

`mintr´6u´12`δ, r´5u´1`δu

178

Applying the operator D‹2 to the above and using (9.40), (9.35) and (9.36) we obtain

2 pF qρD‹2D‹1p´ρ, σq “ p´3ρ` 2 pF qρ2qD‹2D‹1p ˇpF qρ, ˇpF qσq ` 2 pF qρ2

pκD‹2 pF qβ ´ κD‹2 pF qβq


“ p´3ρ` 2 pF qρ2q

ˆ

´1

2pF qρ

`

κpχ` κpχ˘


˙

`2 pF qρ2pκp pF qρpχ`Opr´2u´1`δ

qq ´ κp´ pF qρpχ`mintr´3´δu´12`δ, r´2´δu´1`δuqq


which gives (9.41).

Using (4.25), we express κ in terms of pχ and pχ. Applying the operator D‹2 to

(4.25) and using (9.39), (9.38) and (9.36), we have

D‹2D‹1pκ, 0q “ ´2D‹2D2pχ´ κD‹2ζ ` 2D‹2β ´ 2 pF qρD‹2 pF qβ

“ ´2D‹2D2pχ

´κpr

ˆ

D‹2D2pχ`ˆ

´3

2ρ` pF qρ2

˙

pχ

˙

`1

2rD‹2D‹1pκ, 0q

`mintr´3´δu´12`δ, r´2´δu´1`δuq `Opr´2u´1`δ

qq

which gives (9.42).

We now obtain control over D‹2D‹1pκ, 0q, pχ and pχ by using transport equations.

1. Commuting (4.32) with D‹2D‹1 we obtain

∇ 4pD‹2D‹1pκ, 0qq ` 2κD‹2D‹1pκ, 0q “ 0

179

Together with condition (8.8), this implies

D‹2D‹1pκ, 0q “ 0 for all u ě u0 and r ą rH (9.43)

2. From equation (4.18) and the estimate (6.12) for α we have

∇ 4pχ` κpχ “ Opr´3´δu´12`δ, r´2´δu´1`δq


|pχ| À mintr´2´δu´12`δ, r´1´δu´1`δu for all u ě u0 and r ą rH (9.44)

3. Using (9.39) we deduce

D‹2ζ “ mintr´3´δu´12`δ, r´2´δu´1`δu for all u ě u0 and r ą rH

4. From equation (4.20) and the previous estimates we obtain

∇ 4pχ`1

2κpχ “ Opr´3´δu´12`δ, r´2´δu´1`δ

q


|pχ| À r´1u´1`δ for all u ě u0 and r ą rH (9.45)

5. The decay obtained for D‹2D‹1pκ, 0q, pχ and pχ allows to deduce the following

180

where we obtain decay for K using Gauss equation (4.41).

9.1.4 The terms involved in the e3 direction

We derive here boundedness and decay for the terms involved in the e3 directions,

i.e. for η, ξ and ω.

1. Restricting (4.42) to C0 implies |η| À u´1`δ along C0. Applying Lemma 9.1.0.1

to (4.24) with p “ 1 and q “ 2` δ we obtain

|η| À r´1u´1`δ for all u ě u0 and r ą rH (9.57)

2. Restricting (4.33) and (4.35) to C0 we obtain |ω, Ω, ξ| À u´1`δ. Applying

Lemma 9.1.0.1 to (4.36) with p “ 0 and q “ 3` δ we obtain

|ω| À u´1`δ for all u ě u0 and r ą rH (9.58)

3. Applying Lemma 9.1.0.1 to (4.10) with p “ 1 and q “ 1 we obtain

|ξ| À u´1`δ for all u ě u0 and r ą rH (9.59)

Remark 9.1.1. Observe that the decays hereby obtained are significantly worse than

the optimal decay we are aiming to prove, as stated in Theorem 10.1.1. In particular,

ξ and ω do not present decay in r. Because of this loss of decay in integrating the

equations from the initial data for the quantities ω, ξ and Ω, we will use a different

approach through the SU,R-normalization to prove the optimal decay. From the ini-

tial data normalization we will make use of the boundedness of the solution to apply

Theorem 8.4.1 later.

182

9.1.5 The metric coefficients

We derive here decay for the metric coefficients g, b, Ω and ς.

1. By condition (8.7) and (9.3), equation (4.8) restricted to C0 reads

∇ 3 g “ 0

Using condition (8.10), and integrating the above along C0, we obtain that

g “ 0 on C0. By integrating (4.7) from C0 we obtain

|g| À u´1`δ for all u ě u0 and r ą rH

2. Using (4.5) and the estimate (9.56) we can estimate the projection to the l ě 2

spherical harmonics of trγg:

|D‹2D‹1ptrγg, 0q| À r´2u´1`δ for all u ě u0 and r ą rH (9.60)

On the other hand, integrating (4.16) along C0 and using condition (8.9) we

obtain

|trγgl“1| À u´1`δ along C0

Consequently we have that |trγg| À u´1`δ along C0. Integrating (4.15) we then

obtain

|trγg| À r2u´1`δ for all u ě u0 and r ą rH (9.61)

183

3. Condition (9.4) and the decay for σl“1 implies that |curl bl“1| À u´1`δ along C0.

Combining this with conditions (9.3) and standard elliptic estimates we obtain

|b| À u´1`δ along C0. Applying Lemma 9.1.0.1 to (4.11) with p “ ´1 and q “ 0,

we obtain

|b| À ru´1`δ for all u ě u0 and r ą rH (9.62)

4. Equation (4.9) implies

|ς| À u´1`δ for all u ě u0 and r ą rH


|Ω| À ru´1`δ for all u ě u0 and r ą rH (9.63)

Remark 9.1.2. Observe that the decays obtained for the metric coefficients are also

significantly worse than the optimal decay we are aiming to prove.

In summary, with the use of the initial data normalization we obtain the following

184

decay for the components of the solution S id,K :

|α| ` |β| ď C mintr´3´δu´12`δ, r´2´δu´1`δu

|pχ| ` |ζ| ` | pF qβ| ď C mintr´2´δu´12`δ, r´1´δu´1`δu

|ρ| ` |σ| ` |β| ` |K| ď Cr´2u´1`δ

|pχ| ` | pF qβ| ` |η| ` | ˇpF qρ| ` | ˇpF qσ| ` |κ| ď Cr´1u´1`δ

|g| ` |ξ| ` |ω| ` |ς| ď Cu´1`δ

|b| ` |Ω| ď Cru´1`δ

|trγg| ď Cr2u´1`δ

(9.64)

The growth in r and the non optimal decay for many components forces us to consider

a different normalization in order to obtain the optimal decay stated in Theorem

10.1.1.

9.2 Decay of the pure gauge solution GU,R

In the previous section, we proved that the solution S id,K is bounded in the entire

exterior region. Fix U , R with R " U and R " 3M . By Theorem 8.4.1, we can

associate to S id,K a SU,R-normalized solution

S U,R :“ S id,K´ G U,R

Observe that according to Theorem 8.4.1, the pure gauge solution G U,R is supported

in l ě 1 spherical harmonics, therefore the projection to the l “ 0 spherical harmonics

of S U,R still vanishes. Moreover, the change of gauge does not modify the curl part

of the solution, which is gauge-invariant. Therefore the estimates obtained for that

185

part are not affected.

We derive here decay statements for the functions h, h, a, q1, q2 constructed in the

proof of Theorem 8.4.1.

From (8.34), we have on IU,R, using (9.55)

EpD‹2D‹1pz, 0qq “ κD‹2D‹1pκrS ids, 0q ` κD‹2D‹1pκrS id

s, 0q

À r´4u´1`δ

By Lemma 8.4.0.1, we have on IU,R

|D‹2D‹1pz, 0q| À r´2u´1`δ

Similarly from (8.35) and (8.36), we obtain

|D‹2D‹1pa, 0q| À r´2u´1`δ

|D‹2D‹1pz, 0q| À r´2u´1`δ

which implies on IU,R:

|D‹2D‹1ph, 0q| À r´1u´1`δ|D‹2D‹1ph, 0q| À r´1u´1`δ (9.65)

Integrating (5.6), (5.5) and (5.7) we see that those bounds hold in the whole spacetime.

Similarly for the projection to the l “ 1 spherical harmonics. This implies decay for

all the components of the gauge solution, at a rate which is consistent with the decay

for S id,K .

186

Chapter 10

Proof of linear stability: decay

In this chapter, we prove decay in r and u of a linear perturbation of Reissner-

Nordstrom spacetime S to the sum of a pure gauge solution and a linearized Kerr-

Newman solution.

In Section 10.1 we state the theorem and the exact decay for each component, and

we give an outline of the proof. In Section 10.2 we prove decay of the solution along

the null hypersurface IU,R and finally in Section 10.3 we prove the optimal decay as

stated in the Theorem making use of the SU,R-normalization.

10.1 Statement of the theorem and outline of the

proof

We summarize the statement of linear stability in the following theorem.

Theorem 10.1.1. Let S be a linear gravitational and electromagnetic perturbation

around Reissner-Nordstrom spacetime pM, gM,Qq, with |Q| !M , arising from regular

asymptotically flat initial data. Then, on the exterior of pM, gM,Qq, S decays inverse

187

polynomially to a linearized Kerr-Newman solution K , after adding a pure gauge

solution G which can itself be estimated by the size of the data. In particular,

S ´ G ´K

verifies the following pointwise decay in u and r:

|α| ` |β| ď C mintr´3´δu´12`δ, r´2´δu´1`δu

|ρ| ` |σ| ` |K| ď C mintr´3u´12`δ, r´2u´1`δu

|pF qβ| ď C mintr´2´δu´12`δ, r´1´δu´1`δ

u

|pχ| ` |ζ| ` | ˇpF qρ| ` | ˇpF qσ| ` |κ| ď C mintr´2u´12`δ, r´1u´1`δu

|g| ` |trγg| ď C mintr´1u´12`δ, u´1`δu

and

|β| ď Cr´2u´1`δ

|pχ| ` | pF qβ| ` |η| ` |ξ| ` |ω| ď Cr´1u´1`δ

|ς| ` |Ω| ` |b| ď Cu´1`δ

where C depends on some norms of initial data.

Moreover, the projection to the l “ 0 spherical harmonics of S ´G ´K vanishes

(i.e.(1)

κ “(1)

κ “(1)

ω “(1)

ρ “(1)

pF qρ “(1)

pF qσ “(1)

Ω “ 0).

We outline here the proof of Theorem 10.1.1.

1. SU,R-normalization: By Proposition 9.1.0.1, we have that the solution S id,K

as defined in (9.1) is bounded in the entire exterior region. Fix U , R with R " U

and R " 3M . By Theorem 8.4.1, we can associate to S id,K a SU,R-normalized

188

solution

S U,R :“ S id,K´ G U,R

“ S ´ pG id` G U,R

q ´K id

Observe that according to Theorem 8.4.1, the pure gauge solution G U,R is sup-

ported in l ě 1 spherical harmonics, therefore the projection to the l “ 0

spherical harmonics of S U,R still vanishes.

2. Decay along the null hypersurface IU,R: By definition, the solution S U,R

verifies the conditions of SU,R-normalized solution, which hold along the null

hypersurface IU,R. These conditions imply elliptic relations along IU,R which,

together with the elliptic relations implied by the decay of the gauge-invariant

quantities as summarized in Proposition 9.1.2.1 and 9.1.2.1 imply decay for all

quantities along IU,R.

Recall the charge aspect function ν (defined in (8.11)) and the mass-charge

aspect function µ (defined in (8.12)). The gauge condition for the l “ 1 spherical

harmonics along the null hypersurface IU,R imposes the vanishing of νl“1. Such

condition implies a better decay for µl“1 along the null hypersurface IU,R. In

particular, the quantity µl“1, which in principle would not be bounded in r

along IU,R, is implied to be bounded by the condition of ν, giving

µl“1 À u´1`δ along the null hypersurface IU,R (10.1)

On the other hand, for the projection to the l ě 2 spherical harmonics, the

gauge conditions impose the vanishing of µlě2 along the null hypersurface IU,R.

Finally, the conditions κ “ 0 is necessary to obtain decay in r for the quantities

involved in the e3 direction, namely ω and ξ. Observe that the improved decay

of µl“1 (10.1) is necessary to obtain the decay in r for ωl“1 and ξl“1

. This

189

decay is crucial to improve the result obtained in the previous chapter about

boundedness.

We show decay in u and r for all components of S U,R along the null hypersurface

IU,R. The case l “ 1 and l ě 2 spherical harmonics will be treated separately.

This is done in Section 10.2.

3. Optimal decay in r and u: Once obtained the decay for all the components

along IU,R, we integrate transport equations from IU,R towards the past to

obtain decay in the past of SU,R. We derive the optimal decay in r as given by

the Theorem, with u´12`δ decay. The case l “ 1 and l ě 2 spherical harmonics

will be treated separately. This is done in Section 10.3.2. Obtaining the optimal

decay in u is more delicate, because only one transport equation as given in the

Einstein-Maxwell equations is integrable from IU,R to obtain decay in u´1`δ,

i.e. the equation for κ. Imposing the vanishing of κ at the hypersurface IU,R

then implies the vanishing of it everywhere. This condition by itself is not

enough to obtain optimal decay for all the remaining quantities. It is therefore

crucial to make use of quantities which verify an integrable transport equations

along the e4 direction.

The charge aspect function ν, the mass-charge aspect function µ and a new

quantity Ξ which is a 2-tensor (defined in Lemma 10.3.3.2) satisfy the following:

∇ 4pνl“1q “ 0 (10.2)

∇ 4pµq “ Opr´1´δu´1`δq (10.3)

∇ 4pΞq “ Opr´1´δu´1`δq (10.4)

which all have integrable right hand side.

190

The first two equations will be used in the projection to the l “ 1 spherical

harmonics and the last two equations in the projection to the l ě 2 spherical

harmonics. Observe that the equation for µ will be used differently in the two

cases: indeed in the projection to the l “ 1 spherical harmonics we do not need

to impose the vanishing of µ along IU,R, while for the projection to the l ě 2

spherical harmonics we do. We outline the procedure in each case.

• The projection to the l “ 1 spherical harmonics: Equation (10.2)

together with the gauge condition νl“1 “ 0 implies the vanishing of νl“1

everywhere in the spacetime. The improved decay for µl“1 (10.1) allows to

make use of equation (10.3) to transport the decay for µl“1 in the whole

spacetime exterior. The above implies optimal decay for all the quantities

in the exterior.

• The projection to the l ě 2 spherical harmonics: Equation (10.3)

together with the gauge condition µlě2 “ 0 implies the decay of µlě2

everywhere in the spacetime. In particular, in this part of the solution the

quantity µ plays the same role as ν in the projection to the l “ 1 spherical

harmonics. The new quantity Ξ, only supported in l ě 2 modes, has

the property that is bounded along the null hypersurface IU,R. Therefore

equation (10.4) implies decay for Ξ everywhere, and this is enough to

obtain optimal decay for all the components. In this part of the solution, Ξ

plays the same role as µ in the projection to the l “ 1 spherical harmonics.

Finally, the quantities involved in the e3 direction can be directly integrated

from the null hypersurface IU,R, and now obtain optimal decay.

Observe that the estimates here derived do not depend on U or R, i.e. they

do not depend on the initial far-away sphere SU,R. Therefore the sphere can be

191

taken to be arbitrarily far away, and this implies the derived estimates hold in

the whole exterior region of Reissner-Nordstrom.

In conclusion, the pure gauge solution G “ G id`G U,R and the linearized Kerr-

Newman solution K “ K id as defined above verify the estimates for S´G´K

given by Theorem 10.1.1.

10.2 Decay of the solution along the null hyper-

surface IU,R

The aim of this subsection is to obtain decay in u and r for all components of S U,R on

IU,R for all the curvature components. The case l “ 1 and l ě 2 spherical harmonics

will be treated separately.


Recall conditions (8.13), (8.14), (8.15) verified by the solution SU,R.

In this subsection we show how the above conditions imply the decay for every

component in the projection to the l “ 1 mode along IU,R. We combine the relations

summarized in Proposition 9.1.2.1 to the above gauge conditions. We first compute

192

νl“1. We have, using (9.9) and (9.12),

νl“1 “ r4ppdiv ζql“1 ` 2 pF qρ ˇpF qρq

“ r4p1

rκl“1 ` r

ˆ

3ρ

2 pF qρ´

pF qρ

˙


q

` 2 pF qρp1


1


´1u´1`δqqq

“ r3κl“1 ` r5

ˆ

3ρ

2 pF qρ´

pF qρ`1

2pF qρrκ

˙

pdiv pF qβql“1 ´ r5 pF qρpdiv pF qβql“1

`Opr1´δu´12`δ, r2´δu´1`δq

(10.5)

Conditions (8.13), (8.14) imply, using (9.10)

pdiv pF qβql“1 “1

2rκpdiv pF qβql“1 `Opr

´2u´1`δq (10.6)

On the other hand, the relation (10.5) resticted to I U,R implies

ˆ

3ρ

2 pF qρ´

pF qρ`1

2pF qρrκ

˙

pdiv pF qβql“1 ´pF qρpdiv pF qβql“1

“ Opr´4´δu´12`δ, r´3´δu´1`δq

(10.7)

Using (10.6), we finally obtain

ˆ

3ρ

2 pF qρ´

pF qρ

˙

pdiv pF qβql“1 “ Opr´4´δu´12`δ, r´3´δu´1`δq

which therefore implies

|pdiv pF qβql“1| À mintr´3´δu´12`δ, r´2´δu´1`δu (10.8)

193

Proposition 9.1.2.1 then implies on IU,R:

|pdiv pF qβql“1| À r´2u´1`δ (10.9)

|pdiv βql“1| À mintr´4´δu´12`δ, r´3´δu´1`δu (10.10)

|pdiv βql“1| À r´3u´1`δ (10.11)

|ˇpF qρl“1| À r´1u´1`δ (10.12)

Projecting (A.2) to the l “ 1 spherical harmonics and making use of the fact that

along IU,R r " u, we obtain

|ˇpF qρl“1| À r´2u´12`δ (10.13)

Using the definition of ν and condition (8.15) we observe that div ζl“1 “ ´2 pF qρ ˇpF qρl“1

along I U,R, and therefore (10.12) implies the enhanced estimate for div ζ:

|pdiv ζql“1| À r´3u´1`δ

Using Gauss equation (4.41) and conditions (8.13) and (8.14) we observe that ρl“1 “

2 pF qρ ˇpF qρl“1 and therefore (10.12) implies the enhanced estimate for ρ:

|ρl“1| À r´3u´1`δ (10.14)

The above then implies a better decay than expected for µ at I U,R. Indeed,

|µl“1| “ |r3´

pdiv ζql“1 ` ρl“1 ´ 4 pF qρ ˇpF qρl“1

¯

´ 2r4 pF qρpdiv pF qβql“1|

À r3`

r´3u´1`δ˘

` r4r´2r´2´δu´1`δÀ u´1`δ

(10.15)

194

10.2.2 The projection to the l ě 2 modes

Recall conditions (8.17), (8.18) and (8.19) verified by SU,R on IU,R.

We combine the relations summarized in Proposition 9.1.3.1 to the above gauge

conditions. In order to do so, we first compute D‹2D‹1pµ, 0q. We have

D‹2D‹1pµ, 0q “ r3D‹2D‹1D1ζ ´ 2r4 pF qρD‹2D‹1D1 pF qβ ` r3D‹2D‹1pρ, σq

´4r3 pF qρD‹2D‹1p ˇpF qρ, ˇpF qσq

“ r3p2D‹2D2 ` 2Kq D‹2ζ ´ 2r4 pF qρ p2D‹2D2 ` 2Kq D‹2 pF qβ

`r3D‹2D‹1pρ, σq ´ 4r3 pF qρD‹2D‹1p ˇpF qρ, ˇpF qσq

where we used (8.33). Using relations (9.40) and (9.41), we obtain

D‹2D‹1pµ, 0q “ r3p2D‹2D2 ` 2Kq D‹2ζ ´ 2r4 pF qρ p2D‹2D2 ` 2Kq D‹2 pF qβ

`r3p

ˆ

´3

4ρ´

1

2pF qρ2

˙

`

κpχ` κpχ˘

Òpr´4u´1`δqq

´4r3 pF qρp´1

2pF qρ

`

κpχ` κpχ˘

`mintr´4u´12`δ, r´3u´1`δuq

“ r3p2D‹2D2 ` 2Kq D‹2ζ ´ 2r4 pF qρ p2D‹2D2 ` 2Kq D‹2 pF qβ

`r3

ˆ

´3

4ρ`

3

2pF qρ2

˙

`

κpχ` κpχ˘

Òpr´1u´1`δq

Using relation (9.35), we obtain

D‹2D‹1pµ, 0q “ r3p2D‹2D2 ` 2Kq D‹2ζ ` p2D‹2D2 ` 2Kq p2r4 pF qρ2

pχq

`r3

ˆ

´3

4ρ`

3

2pF qρ2

˙

`

κpχ` κpχ˘

Òpr´1u´1`δq

195

Using relation (9.39), we obtain

D‹2D‹1pµ, 0q “ r4p2D‹2D2 ` 2Kq

ˆ

D‹2D2pχ`ˆ

´3

2ρ` pF qρ2

˙

pχ`1

2D‹2D‹1pκ, 0q

˙

`p2D‹2D2 ` 2Kq p2r4 pF qρ2pχq

`r3

ˆ

´3

4ρ`

3

2pF qρ2

˙

`

κpχ` κpχ˘

Òpr´1u´1`δq

which finally gives

D‹2D‹1pµ, 0q “ r4p2D‹2D2 ` 2Kq pD‹2D2pχq ` r4

p2D‹2D2 ` 2Kq

ˆ

´3

2ρ` 3 pF qρ2

˙

pχ

`1

2r4p2D‹2D2 ` 2Kq D‹2D‹1pκ, 0q ` r3

ˆ

´3

4ρ`

3

2pF qρ2

˙

κpχ

` r3

ˆ

´3

4ρ`

3

2pF qρ2

˙

κpχÒpr´1u´1`δq

(10.16)

which will be used later.

Condition (8.18), (8.17) and relation (9.42) restricted to IU,R imply

2D‹2D2pχ``

´3ρ` 2 pF qρ2˘

pχ “ ´r

2κ`

2D‹2D2pχ``

´3ρ` 2 pF qρ2˘

pχ˘

Òpr´3u´1`δq

Recall the operator E “ 2D‹2D2 ´ 3ρ` 2 pF qρ2, then the above relation becomes

E´

pχ`r

2κpχ

¯

“ Opr´3u´1`δq (10.17)

Applying Lemma 8.4.0.1 to (10.17) we obtain

pχ “ ´r

2κpχÒpr´1u´1`δ

q (10.18)

196

On the other hand, relation (10.16) restricted to IU,R becomes:

r2p2D‹2D2 ` 2Kq pD‹2D2pχq ` r2

p2D‹2D2 ` 2Kq

ˆ

´3

2ρ` 3 pF qρ2

˙

pχ

`r

ˆ

´3

4ρ`

3

2pF qρ2

˙

κpχ`

ˆ

´3

2ρ` 3 pF qρ2

˙

pχ “ Opr´3u´1`δq

Applying (10.18), we finally obtain on IU,R:

p2D‹2D2 ` 2Kq`

2D‹2D2pχ``

´3ρ` 6 pF qρ2˘

pχ˘

“ Opr´5u´1`δq

Recall that 2D‹2D2 ` 2K “ D‹1D1, therefore the standard Poincare inequality implies

2D‹2D2pχ``

´3ρ` 6 pF qρ2˘

pχ À Opr´3u´1`δq (10.19)

A slight modification of Lemma 8.4.0.1 gives an identical Poincare inequality applied

to the operator 2D‹2D2pχ``

´3ρ` 6 pF qρ2˘

which finally implies

|pχ| À Opr´1u´1`δq (10.20)

Relation (10.18) implies

|pχ| À Opr´1u´1`δq (10.21)

197

Proposition 9.1.3.1 implies on IU,R, using that r " u:

|D‹2 pF qβ| À mintr´3´δu´12`δ, r´2´δu´1`δu (10.22)

|D‹2 pF qβ| À Opr´2u´1`δq (10.23)

|D‹2β| À mintr´4´δu´12`δ, r´3´δu´1`δu (10.24)

|D‹2β| À Opr´3u´1`δq (10.25)

|D‹2ζ| À mintr´3´δu´12`δ, r´2´δu´1`δu (10.26)

|D‹2D‹1p ˇpF qρ, ˇpF qσq| À mintr´4u´12`δ, r´3u´1`δu (10.27)

|D‹2D‹1p´ρ, σq| À mintr´5u´12`δ, r´4u´1`δu (10.28)

|D‹2D‹1pK, 0q| À mintr´5´δu´12`δ, r´4u´1`δu (10.29)

Using the above in Codazzi equation (4.26) we finally have on IU,R

|pχ| À Opr´2u´12`δq (10.30)


We obtain here decay along IU,R for the quantities η, ξ, ω.

Conditions (8.13), (8.14) and (8.15) imply ∇ 3κl“1 “ κl“1 “ 0, ∇ 3κl“1 “ κl“1 “ 0

and ∇ 3νl“1 “ νl“1 “ 0 on IU,R. Therefore, using (4.33), (4.35) and (A.5) projected

to the l “ 1 spherical harmonics we have

0 “1

2κ2Ωl“1 ` 2κωl“1 ` 2div ηl“1 ` 2ρl“1, (10.31)

0 “ ´2κωl“1 ` 2div ξl“1`

ˆ

1

2κκ´ 2ρ

˙

Ωl“1 (10.32)

0 “4

r2ωl“1 `

ˆ

1

2κ` 2ω

˙

div pζl“1 ´ ηl“1q `1

2κdiv ξ

l“1` 2 pF qρ2κΩl“1(10.33)

198

The projection of (4.10) to the l “ 1 spherical harmonics implies 2r2

Ωl“1 “ div ξl“1`

Ωpdiv ηl“1 ´ div ζl“1q, therefore relations (10.31) and (10.32) simplify to

´2κωl“1 “ pdiv ξl“1` Ωdiv ηl“1q ` 2div ηl“1 Òpr

´3u´1`δq (10.34)

2κωl“1 “ 2div ξl“1` r

ˆ

1

2κ´ ρ

˙

pdiv ξl“1` Ωdiv ηl“1q Òpr

´3u´1`δq(10.35)

because of the enhanced estimate for div ζl“1 and ρl“1.

Multiplying the first equation by κ and the second equation by κ and summing

the two we obtain

0 “

ˆ

4

r` 2κ´ 2rρ

˙

div ξl“1`

ˆ

4

r` 2κ´ 2rρ

˙

Ωdiv ηl“1 Òpr´4u´1`δ

q

Observe that 4r` 2κ´ 2rρ “ 12M

r2´

8Q2

r2, therefore the above gives

|div ξl“1` Ωdiv ηl“1| À Opr´2u´1`δ

q

Using (4.10) we obtain

|Ω| À Opu´1`δq (10.36)

Multiplying (10.34) by r`

12κ´ ρ

˘

and subtracting (10.35) we obtain

p´4κ` 4rρq ωl“1 “ 2r

ˆ

1

2κ´ rρ

˙

div ηl“1 ´ 2div ξl“1Òpr´3u´1`δ

q(10.37)

Relation (10.33) simplifies, using (10.36) and enhanced estimate for div ζ to

4

r2ωl“1 “

ˆ

1

2κ` 2ω

˙

div ηl“1 ´1

2κdiv ξ

l“1Òpr´4u´1`δ

q

199

Recall that 2ω “ ´rρ, so it can be written as

8

rωl“1 “ 2r

ˆ

1

2κ´ rρ

˙

div ηl“1 ´ 2div ξl“1Òpr´3u´1`δ

q (10.38)

Subtracting (10.38) from (10.37) we then obtain

ˆ

´4κ` 4rρ´8

r

˙

ωl“1 “ Opr´3u´1`δq

and since ´4κ` 4rρ´ 8r“ ´24M

r2`

16Q2

r3we obtain

|ωl“1| À Opr´1u´1`δq

Observe that the enhanced estimates obtained for div ζl“1 and ρ along IU,R allowed

us to obtain the optimal decay for ω, ξ and div ηl“1 and compensate for the loss of a

power of r in the degenerate Poincare inequality above.

Relation (10.35) simplifies to

2κωl“1 “ 2div ξl“1Òpr´2u´1`δ

q

“ ´2Ωdiv ηl“1 Òpr´2u´1`δ

q

which gives

ωl“1 “ ´r

2div ηl“1 Òpr

´1u´1`δq

Using the above we obtain

|div ηl“1, div ξl“1| À Opr´1u´1`δ

q

200

This gives all the desired decay for the projection to the l “ 1 spherical harmonics of

ξ, η, ω and Ω.

Conditions (8.17), (8.18) and (8.19) imply ∇ 3D‹2D‹1pκ, 0q “ D‹2D‹1pκ, 0q “ 0, and

∇ 3D‹2D‹1pκ, 0q “ D‹2D‹1pκ, 0q “ 0 and ∇ 3D‹2D‹1pµ, 0q “ D‹2D‹1pµ, 0q “ 0 on IU,R. There-

fore, using (4.33), (4.35) and Lemma A.2.2.1 commuted with D‹2D‹1 along IU,R we

obtain the following relations:

0 “1

2κ2D‹2D‹1pΩ, 0q ` 2κD‹2D‹1pω, 0q ` 2D‹2D‹1D1η ` 2D‹2D‹1pρ,´σq, (10.39)

0 “ ´2κD‹2D‹1pω, 0q ` 2D‹2D‹1D1ξ `ˆ

1

2κκ´ 2ρ

˙

D‹2D‹1pΩ, 0q (10.40)

and

0 “ 2D‹2D‹1D1D‹1pω, 0q `ˆ

1

2κ` 2ω

˙

D‹2D‹1D1pζ ´ ηq `1

2κD‹2D‹1D1ξ ´ 2D‹2D‹1D1β

` 2 pF qρD‹2D‹1D1 pF qβ ´ˆ

3

2ρ´ 3 pF qρ2

˙

p´κD‹2D‹1pΩ, 0qq ´ 4ωr pF qρD‹2D‹1D1 pF qβ

` 2r pF qρD‹2D‹1D1D‹1p ˇpF qρ, ˇpF qσq ´ 4r pF qρ2D‹2D‹1D1η

(10.41)

Taking into account the estimates already obtained above we can simplify (10.39)

and (10.41) as

1

2κ2D‹2D‹1pΩ, 0q ` 2κD‹2D‹1pω, 0q ` 2D‹2D‹1D1η À Opr´4u´1`δ

q (10.42)

and

2D‹2D‹1D1D‹1pω, 0q ´ˆ

1

2κ` 2ω ` 4r pF qρ2

˙

D‹2D‹1D1η `1

2κD‹2D‹1D1ξ

`

ˆ

3

2ρ´ 3 pF qρ2

˙

κD‹2D‹1pΩ, 0q À Opr´5u´1`δq

(10.43)

201

Using (4.10), relation (10.40) simplifies to

2κD‹2D‹1pω, 0q “ 2D‹2D‹1D1ξ `ˆ

1

2κκ´ 2ρ

˙

D‹2pξ ` Ωηq `Opr´4u´1`δq(10.44)

Again using (4.10), relation (10.42) simplifies to

1

2κ2D‹2pξ ` Ωηq ` 2κD‹2D‹1pω, 0q ` 2D‹2D‹1D1η À Opr´4u´1`δ

q

Using (10.44) to substitute in the above relation upon multiplying by κ, we obtain

κ`

2 p2D‹2D2 ` 2Kq D‹2ξ ` pκκ´ 2ρq D‹2ξ˘

` 2κ p2D‹2D2 ` 2Kq D‹2η

``

κ2κ´ 2κρ˘

D‹2pΩηq À Opr´5u´1`δq

where we used (8.33).. Recalling that κ “ κΩ, we obtain

p4D‹2D2 ` 4K ` κκ´ 2ρq D‹2pξ ` Ωηq À Opr´4u´1`δq

Using Gauss equation the above relation gives

EpD‹2ξ ` ΩD‹2ηq À Opr´4u´1`δq

where E is the operator defined in Lemma 8.4.0.1. By Lemma 8.4.0.1, we obtain

D‹2ξ ` ΩD‹2η À Opr´2u´1`δq

Using (4.10), we obtain

|D‹2D‹1pΩ, 0q| À Opr´2u´1`δq (10.45)

202

Relation (10.44) simplifies to

2κD‹2D‹1pω, 0q “ 2D‹2D‹1D1ξ Òpr´4u´1`δq

“ 2 p2D‹2D2 ` 2Kq D‹2ξ Òpr´4u´1`δq

“ ´2Ω p2D‹2D2 ` 2Kq D‹2η Òpr´4u´1`δq

which gives

κD‹2D‹1pω, 0q “ ´ p2D‹2D2 ` 2Kq D‹2η Òpr´4u´1`δq

Using the above to simplify (10.43), we finally obtain

2 p2D‹2D2 ` 2KqκD‹2D‹1pω, 0q ´`

κκ` 2ωκ` 4rκ pF qρ2˘

p2D‹2D2 ` 2Kq D‹2η

“ ´`

4D‹2D2 ` 4K ` κκ` 2ωκ` 8 pF qρ2˘

p2D‹2D2 ` 2Kq D‹2η

“ ´2`

2D‹2D2 ´ 3ρ` 6 pF qρ2˘

p2D‹2D2 ` 2Kq D‹2η À Opr´6u´1`δq

where we recall that 2ω “ ´rρ. The above operator is then a slight modification of

the operator E , for which a Poincare inequality holds. We therefore have

p2D‹2D2 ` 2Kq D‹2η À Opr´4u´1`δq

The standard Poincare inequality applied to the above then gives

|D‹2η| À Opr´2u´1`δq

203

The above relations imply

|D‹2ξ| À Opr´2u´1`δq

|D‹2D‹1pω, 0q| À Opr´3u´1`δq

Combining the above with the estimates obtained for the projection to the l “ 1

mode we obtain

|ξ| À r´1u´1`δ along IU,R (10.46)

|η| À r´1u´1`δ along IU,R (10.47)

|ω| À r´1u´1`δ along IU,R (10.48)


We derive here decay along IU,R for the metric coefficients g, b and ς.

1. By condition (8.22) and (8.20), integrating equation (4.8) along IU,R gives

|g| À mintr´1u´12`δ, u´1`δu along IU,R (10.49)

2. Using (4.5) and the estimate (10.29) we can estimate the projection to the l ě 2

spherical harmonics of trγg:

|D‹2D‹1ptrγg, 0q| À mintr´3´δu´12`δ, r´2u´1`δu (10.50)

On the other hand, integrating the projection to the l “ 1 spherical harmonics

204

of (4.16) along IU,R and using conditions (8.21) and (8.16) we obtain

|trγgl“1| À mintr´1´δu´12`δ, u´1`δ

u

Consequently we have

|trγg| À mintr´1´δu´12`δ, u´1`δu along IU,R (10.51)

3. Conditions (8.16) and (8.20) imply that b “ 0 along IU,R.


|ς| À u´1`δ along IU,R

10.3 Decay of the solution S U,R in the exterior

Using the decay in u and r along the null hypersurface IU,R as obtained in the

previous section, we will transport it to the past of it using transport equations along

the e4 direction. We summarize the standard computation involved in the integration

along the e4 direction in the following Lemma, where we fix r1 ą rH, very close to

rH.

Lemma 10.3.0.1. If f verifies the transport equation

∇ 4f `p

2κf “ F

205

and f and F satisfy the following estimates:

|f | À r´p´q1u´12`δ on IU,R (10.52)

|F | À r´p´1´q2u´12`δ on tr ě r1u (10.53)

with q1, q2 ě 0, we have for fixed u ă U and any r1 ď r ď R,

|f | À r´p´mintq1,q2uu´12`δ

Proof. According to Lemma 2.3.0.1, the transport equation verified by f is equivalent

to

∇ 4prpfq “ rpF

Using (3.16), the transport equation becomes

Brprpfq “ rpF

Consider now a fixed u ă U . The null hypersurface of fixed u intersects IU,R at a

certain r “ r˚puq in the sphere Su,r˚puq. We now integrate the above equation along

the fixed u hypersurface from the sphere Su,r˚puq on IU,R to the sphere Su,r for any

r1 ď r ď R. We obtain

rpfpu, rq “ r˚puqpfpu, r˚puqq `

ż r

r˚

λpF pu, λqdλ

If condition (10.52) is satisfied, then |fpu, r˚puqq| À r˚puq´p´q1u´12`δ and |F pu, λq| À

206

λ´p´1´q2u´12`δ, which gives

rp|fpu, rq| À r˚puq´q1u´12`δ

`

ż r

r˚

λ´1´q2u´12`δdλ

À r˚puq´q1u´12`δ

` r´q2u´12`δ` r˚puq

´q2u´12`δ

Since by construction r˚puq ě R for every u ă U , and q1 ě 0, we can bound the right

hand side by

rp|fpu, rq| À r´q2u´12`δ`R´mintq1,q2uu´12`δ

Since R ě r we can bound R´mintq1,q2u ď r´mintq1,q2u, and finally obtain an estimate

which is independent of R:

rp|fpu, rq| À r´mintq1,q2uu´12`δ

Diving by rp, we obtain the desired estimate.

10.3.1 The projection to the l “ 1 mode: optimal decay in r

and in u

1. Integrating (4.32) and using condition (8.13) we obtain

κl“1 “ 0 for all u ě u0 and r ě rH (10.54)

2. The projection of (A.4) to the l “ 1 spherical harmonics gives, using (10.54),

∇ 4νl“1 “ r2κl“1 ´ 2r4 pF qρ2κl“1 ` r4div div pχl“1 “ 0

207

Integrating the above from I U,R and using condition (8.15) we obtain

νl“1 “ 0 for all u ě u0 and r ě rH (10.55)

Using the relation implied by ν (10.7) and (10.55) we obtain everywhere in the

spacetime

ˆ

3ρ

2 pF qρ´

pF qρ`1

2pF qρrκ

˙

pdiv pF qβql“1 ´pF qρpdiv pF qβql“1

“ Opr´4´δu´12`δ, r´3´δu´1`δq

(10.56)

3. The projection of (A.6) to the l “ 1 spherical harmonics gives, using (10.54)

∇ 4µl“1 “ Opr´1´δu´1`δq

Consider now a fixed u ă U . As in Lemma 10.3.0.1, we integrate the above

equation along the fixed u hypersurface from the sphere Su,r˚puq on IU,R to the

sphere Su,r for any r0 ď r ď R. We obtain

µl“1 À µl“1pu, r˚puqq `

ż r

r˚

λ´1´δu´1`δdλ

Because of estimate (10.15), we obtain

|µl“1| À u´1`δ`

ż r

r˚

λ´1´δu´1`δdλ

À u´1`δ

208

We have, using (9.9), (9.11) and (9.12),

µl“1 “ r3ppdiv ζql“1 ` ρl“1 ´ 4 pF qρ ˇpF qρl“1q ´ 2r4 pF qρpdiv pF qβql“1

“ r3p1

rκl“1 ` r

ˆ

3ρ

2 pF qρ´

pF qρ

˙


q

`1

4r2κ

ˆ

3ρ

2 pF qρ`

pF qρ

˙


2r

ˆ

3ρ

2 pF qρ`

pF qρ

˙

pdiv pF qβql“1

Òpr´2u´1`δq ´ 4 pF qρp

1


1


´1u´1`δqqq

´ 2r4 pF qρpdiv pF qβql“1

“ r2κl“1 ` r4

ˆ

3ρ

2 pF qρ´ 3 pF qρ

˙

pdiv pF qβql“1

`1

4r5κ

ˆ

3ρ

2 pF qρ´ 3 pF qρ

˙


2r4

ˆ

3ρ

2 pF qρ´ 3 pF qρ

˙

pdiv pF qβql“1

Òpru´1`δq

(10.57)

Using the estimate for µ obtained above we have

pdiv pF qβql“1 “

ˆ

2`1

2rκ

˙

pdiv pF qβql“1 Òpr´2u´1`δ

q (10.58)

Substituting the above into (10.56) we finally obtain

ˆ

3ρ

2 pF qρ´ 3 pF qρ

˙

pdiv pF qβql“1 “ Opr´4´δu´12`δ, r´3´δu´1`δq

which gives

|pdiv pF qβql“1| À Opr´3´δu´12`δ, r´2u´1`δq (10.59)

209

4. Relation (10.58) implies then

|pdiv pF qβql“1| À Opr´2u´1`δq (10.60)

5. The decay obtained for κl“1, pdiv pF qβql“1, pdiv pF qβql“1 allows to deduce the

following decays for all u ě u0 and r ě r1 using Proposition 9.1.2.1:

|pdiv ζql“1| À mintr´3´δu´12`δ, r´2´δu´1`δu (10.61)

|pdiv βql“1| À mintr´4´δu´12`δ, r´3´δu´1`δu (10.62)

|pdiv βql“1| À r´3u´1`δ (10.63)

|κl“1| À r´1u´1`δ (10.64)

|ρl“1| À r´2u´1`δ (10.65)

|ˇpF qρl“1| À r´1u´1`δ (10.66)

6. Using (10.59) and (10.13), we apply Lemma 10.3.0.1 to (4.49) with p “ 2, q1 “ 0

and q2 “ δ and obtain

|ˇpF qρl“1| À r´2u´12`δ (10.67)

Similarly, using (10.59), (10.62), (10.67), (10.14), we apply Lemma 10.3.0.1 to

(4.61) with p “ 3, q1 “ 0 and q2 “ δ and obtain

|ρl“1| À r´3u´12`δ (10.68)

Similarly, using (10.61), (10.68) and condition κl“1 “ 0 on IU,R, and integrating

210

(4.34) we obtain

|κl“1| À r´2u´12`δ (10.69)

10.3.2 The projection to the l ě 2 modes: optimal decay in r

Using the decay in u and r along IU,R, we will transport it to the past of SU,R using

transport equations.

We apply Lemma 10.3.0.1 to obtain optimal decay in r (and decay in u as u´12`δ),

for some of the components.

1. Applying Lemma 10.3.0.1 to (4.32) for f “ D‹2D‹1pκ, 0q, F “ 0, and using

condition (8.17), we obtain

D‹2D‹1pκ, 0q “ 0 on tr1 ď r ď Ru (10.70)

2. Using (10.30) and (6.12), ee apply Lemma 10.3.0.1 to (4.18) for f “ pχ, F “ ´α,

p “ 2, q1 “ 0, q2 “ δ. We obtain

|pχ| À r´2u´12`δ on tr1 ď r ď Ru (10.71)

3. Using (9.39), (9.35) and (9.37) and the above we obtain

|D‹2 pF qβ| À r´3´δu´12`δ (10.72)

|D‹2β| À r´4´δu´12`δ (10.73)

|D‹2ζ| À r´3u´12`δ (10.74)

211

4. Commuting (4.49) by D‹2D‹1 we obtain

∇ 4pD‹2D‹1p ˇpF qρ, ˇpF qσqq ` 2κD‹2D‹1p ˇpF qρ, ˇpF qσq “ D‹2D‹1D1 pF qβ

“ p2D‹2D2 ` 2Kq D‹2 pF qβ

We apply Lemma 10.3.0.1 to the above for f “ D‹2D‹1p ˇpF qρ, ˇpF qσq, p “ 4, q1 “ 0,

q2 “ δ. Using (10.27) we obtain

|D‹2D‹1p ˇpF qρ, ˇpF qσq| À r´4u´12`δ (10.75)

A similar procedure applies to D‹2D‹1p´ρ, σq and D‹2D‹1pκ, 0q. We obtain

|D‹2D‹1pρ, σq| À r´5u´12`δ (10.76)

|D‹2D‹1pκ, 0q| À r´4u´12`δ (10.77)

This completes the proof of the optimal decay in r for the above quantities.

10.3.3 The projection to the l ě 2 modes: optimal decay in

u

We derive now the decay estimates which imply decay of order u´1`δ for all the

curvature components in the linear stability. In order to do so, we make use of two

quantities: the first is µ as introduced in the gauge normalization, and the second

one is a new quantity Ξ. Both these quantities have the property that the transport

in a very good way from IU,R in the e4 direction.

Lemma 10.3.3.1. The mass-charge aspect function µ defined by scalar defined in

212

(8.12) verifies on tr1 ď r ď Ru the following estimate:

|D‹2D‹1pµ, 0q| À r´2´δu´1`δ

Proof. Commuting (A.6) with r2D‹2D‹1 and using (10.70), we obtain

∇ 4pr2D‹2D‹1pµ, 0qq “ Opr´1´δu´1`δ

q

Therefore r2D‹2D‹1pµ, 0q verifies the transport equation

Brpr2D‹2D‹1pµ, 0qq “ Opr´1´δu´1`δ

q

Consider now a fixed u ă U . As in Lemma 10.3.0.1, we integrate the above equation

along the fixed u hypersurface from the sphere Su,r˚puq on IU,R to the sphere Su,r for

any r0 ď r ď R. We obtain

r2D‹2D‹1pµ, 0q À r˚puq2D‹2D‹1pµ, 0qpu, r˚puqq `

ż r

r˚

λ´1´δu´1`δdλ

Because of condition (8.19), we obtain

|r2D‹2D‹1pµ, 0qpu, rq| Àż r

r˚

λ´1´δu´1`δdλ

À r´δu´1`δ

for all r0 ď r ď R, as desired.

In addition to µ we define the following quantity, which has the property that

verifies a good transport equation in the e4 direction.

213

Lemma 10.3.3.2. The traceless symmetric two tensor defined as

Ξ :“

ˆ

κr2` ρr3

´2

3r3 pF qρ2

˙

pχ´ κr2pχ´ 4r2D‹2ζ ` 2r3D‹2β

´2

3r3 pF qρD‹2 pF qβ

(10.78)

verifies on tr1 ď r ď Ru the following estimate:

|Ξ| À u´1`δ

Proof. We compute ∇ 4Ξ.

We start by computing ∇ 4pκr2pχq.

∇ 4pκr2pχq “ ∇ 4pκqr

2pχ` κ∇ 4pr

2qpχ` κr2∇ 4ppχq

“ p´1

2κκ` 2ρqr2

pχ` κκr2pχ` κr2

p´κpχ´ αq

“

ˆ

´1

2κκ` 2ρ

˙

r2pχ´ κr2α

We compute ∇ 4pρr3pχq:

∇ 4pρr3pχq “ ∇ 4pρqr

3pχ` ρ∇ 4pr

3qpχ` ρr3∇ 4ppχq

“ p´3

2κρ´ κ pF qρ2

qr3pχ` ρ

3

2r3κpχ` ρr3

p´κpχ´ αq

“ p´2ρ´ 2 pF qρ2qr2

pχ´ ρr3α

214

We compute ∇ 4p23r3 pF qρ2

pχq.

∇ 4p2

3r3 pF qρ2

pχq “2

3∇ 4pr

3qpF qρ2

pχ`2

3r3∇ 4p

pF qρ2qpχ`

2

3r3 pF qρ2∇ 4ppχq

“ r3κ pF qρ2pχ´

4

3r3κ pF qρ2

pχ`2

3r3 pF qρ2

p´κpχ´ αq

“ ´2r2 pF qρ2pχ´

2

3r3 pF qρ2α

We therefore obtain

∇ 4

ˆˆ

κr2` ρr3

´2

3r3 pF qρ2

˙

pχ

˙

“ ´1

2κκr2

pχ`

ˆ

´κr2´ ρr3

`2

3r3 pF qρ2

˙

α(10.79)

We compute ∇ 4pκr2pχq.

∇ 4pκr2pχq “ ∇ 4pκqr

2pχ` κ∇ 4pr

2qpχ` κr2∇ 4ppχq

“ p´1

2κ2qr2

pχ` κ2r2pχ` κr2

p´1

2κpχ` 2D‹2ζ ´

1

2κpχq

“ ´1

2r2κκpχ` 2κr2D‹2ζ

Putting it together with (10.79), we obtain

∇ 4

ˆˆ

κr2` ρr3

´2

3r3 pF qρ2

˙

pχ´ κr2pχ

˙

“ ´2κr2D‹2ζ

`

ˆ

´κr2´ ρr3

`2

3r3 pF qρ2

˙

α

(10.80)

Commuting (4.22) with rD‹2 we obtain

∇ 4prD‹2ζq ` κrD‹2ζ “ ´rD‹2β ´ pF qρrD‹2 pF qβ

∇ 4prD‹2ζq `1

2κrD‹2ζ “ ´

1

2κrD‹2ζ ´ rD‹2β ´ pF qρrD‹2 pF qβ

215

which can be written as

∇ 4pr2D‹2ζq “ ´

1

2κr2D‹2ζ ´ r2D‹2β ´ pF qρr2D‹2 pF qβ (10.81)

Combining (10.80) and (10.81) we obtain

∇ 4

ˆˆ

κr2` ρr3

´2

3r3 pF qρ2

˙

pχ´ κr2pχ´ 4r2D‹2ζ

˙

“ 4r2D‹2β ` 4 pF qρr2D‹2 pF qβ

`

ˆ

´κr2´ ρr3

`2

3r3 pF qρ2

˙

α

Commuting (4.57) with rD‹2 we obtain

∇ 4prD‹2βq ` 2κrD‹2β “ rD‹2D2α ` pF qρ∇ 4prD‹2 pF qβq

∇ 4prD‹2βq ` κrD‹2β “ ´κrD‹2β ` rD‹2D2α ` pF qρ∇ 4prD‹2 pF qβq


∇ 4pr3D‹2βq “ ´κr3D‹2β ` r3D‹2D2α ` r2 pF qρ∇ 4prD‹2 pF qβq

Since r2 pF qρ “ Q, we obtain

∇ 4pr3D‹2β ´ r3 pF qρD‹2 pF qβq “ ´2r2D‹2β ` r3D‹2D2α (10.82)

Commuting (A.3) with rD‹2 we obtain

∇ 4prD‹2 pF qβq `3

2κrD‹2 pF qβ “ p´3ρ` 2 pF qρ2

q´1

´

∇ 4prD‹2βq ` 3κrD‹2β ´ 2 pF qρrD‹2D2α¯

216

We therefore compute ∇ 4pr3 pF qρD‹2 pF qβq.

∇ 4pr3 pF qρD‹2 pF qβq “ Q∇ 4prD‹2 pF qβq

“ Qp´3

2κrD‹2 pF qβ

`p´3ρ` 2 pF qρ2q´1

´


q

“ ´3r2 pF qρD‹2 pF qβ

`Qp´3ρ` 2 pF qρ2q´1

´


We can finally put together the above computations, and obtain

∇ 4Ξ “

ˆ

´κr2´ ρr3

`2

3r3 pF qρ2

˙

α ` 2pr3D‹2D2αq

`4

3pQp´3ρ` 2 pF qρ2

q´1

´


q

Using the estimate for α and β given by (6.12) and (6.14), we can bound the right

hand side of the above by

|rα| ` |r2β| À r´1´δu´1`δ

We integrate the above equation along the fixed u hypersurface from the sphere

Su,r˚puq on IU,R to the sphere Su,r for any r0 ď r ď R. We obtain

Ξpu, rq À Ξpu, r˚puqq `

ż r

r˚

λ´1´δu´1`δdλ

On IU,R, the quantity Γ verifies the following estimate

|Ξpu, r˚puqq| ď |rpχ| ` |rpχ| ` |r2D‹2ζ| ` |r3D‹2β| ` |r´1D‹2 pF qβ| À u´1`δ

217

where we used (10.20), (10.21), (10.26), (10.24) and (10.22).

Therefore we finally obtain

|Ξpu, rq| À u´1`δ`

ż r

r˚

λ´1´δu´1`δdλ À u´1`δ

for all r0 ď r ď R, as desired.

The decay for the above two quantities implies decay for all the remaining quan-

tities.

We write Ξ using the expressions given by Proposition 9.1.3.1. We have, using

(10.70):

Ξ “

ˆ

κr2` ρr3

´2

3r3 pF qρ2

˙

pχ´ κr2pχ´ 4r2D‹2ζ ` 2r3D‹2β ´

2

3r3 pF qρD‹2 pF qβ

“

ˆ

κr2` ρr3

´2

3r3 pF qρ2

˙

pχ´ κr2pχ´ 4r2

pr

ˆ

D‹2D2pχ`ˆ

´3

2ρ` pF qρ2

˙

pχ

˙

q

`2r3p´

3

2ρpχq ´

2

3r3 pF qρp´ pF qρpχq Òpr´δu´1`δ

q

“`

κr2` 4ρr3

´ 4 pF qρ2r3˘

pχ´ κr2pχ´ 4r3

pD‹2D2pχq Òpr´δu´1`δq

Recalling the estimate for Ξ given by Lemma 10.3.3.2, we can write pχ in terms of pχ:

pχ “ r2

ˆ

´2D‹2D2 `1

4κκ` 2ρ´ 2 pF qρ2

˙

pχÒpr´1u´1`δq

“ r2`

´2D‹2D2 ´K ` ρ´ pF qρ2˘

pχÒpr´1u´1`δq

finally giving

pχ “ r2

ˆ

´2D‹2D2 ´ 2K ´1

4κκ

˙

pχÒpr´1u´1`δq (10.83)

Now recall relation (10.16). We use the expressions given by Proposition 9.1.3.1.

218

We have, using (10.70) and (10.83):

D‹2D‹1pµ, 0q “ r4p2D‹2D2 ` 2Kq pD‹2D2pχq ` r4

p2D‹2D2 ` 2Kq

ˆ

´3

2ρ` 3 pF qρ2

˙

pχ

`r3

ˆ

´3

4ρ`

3

2pF qρ2

˙

κpχ` r4

ˆ

3

2ρ´ 3 pF qρ2

˙ˆ

2D‹2D2 ` 2K `1

4κκ

˙

pχ

`Opr´1u´1`δq

“ r4p2D‹2D2 ` 2Kq pD‹2D2pχq `Opr´1u´1`δ

q

Using the estimate for D‹2D‹1pµ, 0q obtained in Lemma 10.3.3.1, we have

p2D‹2D2 ` 2Kq pD‹2D2pχq “ D‹2D‹1D1D2pχ “ Opr´5u´1`δq

The above operator is clearly coercive. Indeed,

ż

S

pχ ¨ D‹2D‹1D1D2pχ “

ż

S

|D1D2pχ|2 ě1

r4

ż

S

|pχ|2

This gives


Relation (10.83) then implies


219


We finally derive here the decay for the metric coefficients g, b, Ω and ς.


|g| À mintr´1u´12`δ, u´1`δu


|trγg| À mintr´1´δu´12`δ, u´1`δu

3. Integrating (4.11) we obtain

|b| À u´1`δ


|ς| À u´1`δ


|Ω| À u´1`δ

10.3.6 Decay close to the horizon

Recall that we fixed r1 ą rH, very close to rH. As explained in Section 3.2, the Bondi

coordinates pu, r, θ, φq do not cover the boundary of the exterior of the spacetime, and

223

therefore they do not cover the horizon. Moreover, the null frame N is not regular

towards the horizon, while N˚ “ tΩ´1e3,Ωe4u is regular towards the horizon.

Since we obtained the above bounds for the components of the solution up to

tr “ r1u, in order to make sense of the same bounds towards the horizon, we have

to consider the same bounds applied to the right rescaling that makes the above

components regular towards the horizon.

For example, the following quantities are regular towards the horizon:

Ω2α, Ω´2α, Ωpχ, Ω´1pχ, g, ζ, η, Ω´2ξ, Ωβ, Ω´1β,

Ω pF qβ, Ω´1 pF qβ, b

and the estimates derived above imply estimates for the quantities regular towards

the horizon.

224

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230

Appendix A

Explicit computations

We derive in this appendix relations involving the relevant gauge-invariant quantities

which are used in the proof of linear stability.

A.1 Alternative expressions for qF and p

We summarize in the following lemma alternative expressions for qF and p which

differ from their definitions given in Section 6.3.

Lemma A.1.0.1. The following relations hold true:

qF

r3“ ´D‹2D‹1p ˇpF qρ, ˇpF qσq ´

1

2pF qρ

`

κpχ` κpχ˘

, (A.1)

p

r5“ 2 pF qρD‹1p´ρ, σq ` p3ρ´ 2 pF qρ2

qD‹1p ˇpF qρ, ˇpF qσq ` 2 pF qρ2pκ pF qβ ´ κ pF qβq(A.2)

Proof. The gauge-invariant quantity qF is by definition (6.5) given by

qF “1

2rpr2κfq `

1

κr∇ 3pr

2κfq “1

2rpr2κfq `

1

κr

ˆ

r2κ∇ 3pfq `

ˆ

1

2κ´ 2ω

˙

r2κf

˙

“ r3p∇ 3pfq ` pκ´ 2ωq fq

231

Using the definition of f (5.10), we compute

∇ 3pfq ` pκ´ 2ωq f “ ∇ 3pD‹2 pF qβ ` pF qρpχq ` pκ´ 2ωq pD‹2 pF qβ ` pF qρpχq

“ D‹2∇ 3pF qβ ´

1

2κD‹2 pF qβ `∇ 3

pF qρpχ` pF qρ∇ 3pχ

`pκ´ 2ωq pD‹2 pF qβ ` pF qρpχq

and using (4.42) and (4.19), we obtain

∇ 3pfq ` pκ´ 2ωq f “ ´

ˆ

1

2κ´ 2ω

˙

D‹2 pF qβ ´ D‹2D‹1p ˇpF qρ, ˇpF qσq ` 2 pF qρD‹2η

´1

2κD‹2 pF qβ ´ κ pF qρpχ

`pF qρp´

ˆ

1

2κ´ 2ω

˙

pχ´ 2 D‹2η ´1

2κpχq

` pκ´ 2ωq pD‹2 pF qβ ` pF qρpχq

“ ´D‹2D‹1p ˇpF qρ, ˇpF qσq ´1

2pF qρ

`

κpχ` κpχ˘

which proves (A.1).

The gauge-invariant quantity p is by definition (6.5) given by

p “1

2rpr4κβq `

1

κr∇ 3pr

4κβq “1

2rpr4κβq `

1

κrp

3

2κ´ 2ωqr4κβ `

1

κrr4κ∇ 3pβq

“ r5´

∇ 3pβq ` p2κ´ 2ωqβ¯

232

Using the definition of β (5.13), we compute

∇ 3pβq ` p2κ´ 2ωqβ “ ∇ 3p2pF qρβ ´ 3ρ pF qβq ` p2κ´ 2ωqp2 pF qρβ ´ 3ρ pF qβq

“ 2∇ 3ppF qρqβ ` 2 pF qρ∇ 3pβq ´ 3∇ 3pρq

pF qβ ´ 3ρ∇ 3ppF qβq

`p2κ´ 2ωqp2 pF qρβ ´ 3ρ pF qβq

“ ´2κ pF qρβ ` 2 pF qρp´ pκ´ 2ωq β ` D‹1p´ρ, σq ` 3ρζ

`pF qρ

ˆ

´D‹1p ˇpF qρ, ˇpF qσq ´ κ pF qβ ´1

2κ pF qβ

˙

q

´3p´3

2κρ´ κ pF qρ2

qpF qβ ´ 3ρp´

ˆ

1

2κ´ 2ω

˙

pF qβ

´D‹1p ˇpF qρ, ˇpF qσq ` 2 pF qρζq

`p2κ´ 2ωqp2 pF qρβ ´ 3ρ pF qβq

“ 2 pF qρD‹1p´ρ, σq ` p3ρ´ 2 pF qρ2qD‹1p ˇpF qρ, ˇpF qσq

`2 pF qρ2pκ pF qβ ´ κ pF qβq

which proves (A.2).

A.2 Remarkable transport equations

We summarize here some transport equations which are important in the proof of

linear stability.

Lemma A.2.0.1. The following transport equations hold:

∇ 4pF qβ `

3

2κ pF qβ “ p´3ρ` 2 pF qρ2

q´1

´

∇ 4β ` 3κβ ´ 2 pF qρdiv α¯

(A.3)

233

∇ 3pF qβ `

ˆ

3

2κ` 2ω

˙

pF qβ ` 2 pF qρξ “ p´3ρ` 2 pF qρ2q´1

´

∇ 3β ` p3κ` 2ωqβ ` 2 pF qρdiv α¯

Proof. We compute, using Maxwell equations and Bianchi identities:

∇ 4β ` 3κβ “ ∇ 4p2pF qρβ ´ 3ρ pF qβq ` 3κp2 pF qρβ ´ 3ρ pF qβq

“ 2∇ 4pF qρβ ` 2 pF qρ∇ 4β ´ 3∇ 4ρ

pF qβ ´ 3ρ∇ 4pF qβ ` 3κp2 pF qρβ ´ 3ρ pF qβq

“ 2p´κ pF qρqβ ` 2 pF qρp´2κβ ` div α ` pF qρ∇ 4pF qβq

´3p´3

2κρ´ κ pF qρ2

qpF qβ ´ 3ρ∇ 4

pF qβ ` 3κp2 pF qρβ ´ 3ρ pF qβq

“ 2 pF qρdiv α ` p2 pF qρ2´ 3ρqp∇ 4

pF qβ `3

2κ pF qβq

which proves (A.3). Similarly,

∇ 3β ` p3κ` 2ωqβ “ 2∇ 3pF qρβ ` 2 pF qρ∇ 3β ´ 3∇ 3ρ

pF qβ ´ 3ρ∇ 3pF qβ

`p3κ` 2ωqp2 pF qρβ ´ 3ρ pF qβq

“ 2p´κ pF qρqβ ` 2 pF qρp´2κβ ´ 2ωβ ´ div α ´ 3ρξ

`pF qρp∇ 3

pF qβ ` 2ω pF qβ ` 2 pF qρξqq

´3p´3

2κρ´ κ pF qρ2

qpF qβ ´ 3ρ∇ 3

pF qβ ` p3κ` 2ωqp2 pF qρβ ´ 3ρ pF qβq

“ ´2 pF qρdiv α ` p2 pF qρ2´ 3ρqp∇ 3

pF qβ ` p3

2κ` 2ωq pF qβ ` 2 pF qρξq

A.2.1 The charge aspect function

Recall the definition of the the charge aspect function ν in (8.11). We compute in

the following lemma the transport equation for the charge aspect function.

234

Lemma A.2.1.1. The function ν verifies the following transport equation:

∇ 4ν “1

2r4D1D‹1pκ, 0q ´ 2r4 pF qρ2κ` r4div div pχ (A.4)

∇ 3ν “ r4

ˆ

2D1D‹1pω, 0q `´1

2κ` 2ω

˙

div pζ ´ ηq `1

2κdiv ξ ´ div div pχ

´1

2D1D‹1pκ, 0q ´ 2 pF qρ2

`

κ´ κΩ˘

¯

(A.5)

Proof. We compute, using (4.22) and (4.49):

∇ 4

ˆ

ν

r4

˙

“ ∇ 4

´


“ div p∇ 4ζq ´1

2κdiv ζ ` 2p∇ 4

pF qρq ˇpF qρ` 2 pF qρ∇ 4ˇpF qρ

“ div p´κζ ´ β ´ pF qρ pF qβq ´1

2κdiv ζ ´ 2κ pF qρ ˇpF qρ

`2 pF qρp´κ ˇpF qρ´ pF qρκ` div pF qβq

We obtain

∇ 4

ˆ

ν

r4

˙

“ ´div β ` pF qρdiv pF qβ ´3

2κdiv ζ ´ 4κ pF qρ ˇpF qρ´ 2 pF qρ2κ

Taking the divergence of Codazzi equation (4.26), we can write

´div β ` pF qρdiv pF qβ “ div div pχ´1

2κdiv ζ `

1

2D1D‹1pκ, 0q

235

Substituting in the above we finally obtain

∇ 4

ˆ

ν

r4

˙

“ div div pχ´1

2κdiv ζ `

1

2D1D‹1pκ, 0q ´

3

2κdiv ζ ´ 4κ pF qρ ˇpF qρ´ 2 pF qρ2κ

“ ´2κdiv ζ ´ 4κ pF qρ ˇpF qρ` div div pχ`1

2D1D‹1pκ, 0q ´ 2 pF qρ2κ

“ ´2κ

ˆ

µ

r4

˙

` div div pχ`1

2D1D‹1pκ, 0q ´ 2 pF qρ2κ

as desired.

We also compute, using (4.21) and (4.48):

∇ 3

ˆ

ν

r4

˙

“ ∇ 3

´


“ div p∇ 3ζq ´1

2κdiv ζ ` 2p∇ 3

pF qρq ˇpF qρ` 2 pF qρ∇ 3ˇpF qρ

“ div p´

ˆ

1

2κ´ 2ω

˙

ζ ` 2D‹1pω, 0q ´ˆ

1

2κ` 2ω

˙

η `1

2κξ ´ β ´ pF qρ pF qβq

´1

2κdiv ζ ´ 2κ pF qρ ˇpF qρ` 2 pF qρp´κ ˇpF qρ´ pF qρ

`

κ´ κΩ˘

´ div pF qβq

“ ´ pκ´ 2ωq div ζ ` 2D1D‹1pω, 0q ´ˆ

1

2κ` 2ω

˙

div η `1

2κdiv ξ

´div β ` pF qρdiv pF qβ ´ 4κ pF qρ ˇpF qρ´ 2 pF qρ2`

κ´ κΩ˘

Taking the divergence of Codazzi equation (4.25), we can write

´div β ` pF qρdiv pF qβ “ ´div div pχ´1

2κdiv ζ ´

1

2D1D‹1pκ, 0q

236

Substituting in the above, we finally have

∇ 3

ˆ

ν

r4

˙

“ ´

ˆ

3

2κ´ 2ω

˙

div ζ ` 2D1D‹1pω, 0q ´ˆ

1

2κ` 2ω

˙

div η `1

2κdiv ξ

´div div pχ´1

2D1D‹1pκ, 0q ´ 4κ pF qρ ˇpF qρ´ 2 pF qρ2

`

κ´ κΩ˘

“ ´2κdiv ζ ´ 4κ pF qρ ˇpF qρ` 2D1D‹1pω, 0q `ˆ

1

2κ` 2ω

˙

pdiv ζ ´ div ηq

`1

2κdiv ξ ´ div div pχ´

1

2D1D‹1pκ, 0q ´ 2 pF qρ2

`

κ´ κΩ˘

as desired.

A.2.2 The mass-charge aspect function

Recall the definition of the the mass-charge aspect function µ in (8.12). We compute

in the following lemma the transport equation for the mass-charge aspect function.

Lemma A.2.2.1. The function µ verifies the following transport equations:

∇ 4µ “ ´r3

ˆ

3

2ρ´ 7 pF qρ2

˙

κ`Opr´1´δu´1`δq (A.6)

∇ 3µ “ r3´

2D1D‹1pω, 0q `ˆ

1

2κ` 2ω

˙

pdiv ζ ´ div ηq `1

2κdiv ξ ´ 2div β ` 2 pF qρdiv pF qβ

´

ˆ

3

2ρ´ 3 pF qρ2

˙

pκ´ κΩq ´ 4ωr pF qρdiv pF qβ ` 2r pF qρD1D‹1p ˇpF qρ, ˇpF qσq ´ 4r pF qρ2div η

(A.7)

Proof. We compute ∇ 4µ.

Commuting (4.22) with rdiv we obtain

∇ 4prdiv ζq ` κrdiv ζ “ ´rdiv β ´ pF qρrdiv pF qβ

237


∇ 4pr3div ζq “ ´r3div β ´ pF qρr3div pF qβ (A.8)

Equation (4.61) can be written as

∇ 4pr3ρq “ ´r3

ˆ

3

2ρ` pF qρ2

˙

κ´ 4r2 pF qρ ˇpF qρ` r3div β ` r3 pF qρ div pF qβ (A.9)

Equation (4.49) implies

∇ 4ppF qρ ˇpF qρq “ ´κ pF qρ ˇpF qρ` pF qρp´κ ˇpF qρ´ pF qρκ` div pF qβq

“ ´2κ pF qρ ˇpF qρ´ pF qρ2κ` pF qρdiv pF qβ


∇ 4p´4r3 pF qρ ˇpF qρq “ 4r2 pF qρ ˇpF qρ´ 4r3 pF qρdiv pF qβ ` 4r3 pF qρ2κ (A.10)

Commuting (A.3) with div and using estimates for β we obtain

∇ 4div pF qβ ` 2κdiv pF qβ “ Opr´3´δu´1`δq

which implies

∇ 4ppF qρdiv pF qβq “ ´κ pF qρdiv pF qβ ` pF qρp´2κdiv pF qβ `Opr´3´δu´1`δ

qq

“ ´3κ pF qρdiv pF qβ `Opr´5´δu´1`δq

238


∇ 4p´2r4 pF qρdiv pF qβq “ 4r3 pF qρdiv pF qβ Òpr´1´δu´1`δq (A.11)

Summing (A.8), (A.9), (A.10), (A.11), we obtain

∇ 4µ “ ´r3div β ´ pF qρr3div pF qβ ´ 4r2 pF qρ ˇpF qρ` r3div β ` r3 pF qρ div pF qβ

`4r2 pF qρ ˇpF qρ´ 4r3 pF qρdiv pF qβ ` 4r3 pF qρ2κ` 4r3 pF qρdiv pF qβ ´ r3

ˆ

3

2ρ´ 3 pF qρ2

˙

κ

Òpr´1´δu´1`δq

“ ´r3

ˆ

3

2ρ´ 7 pF qρ2

˙

κÒpr´1´δu´1`δq

Using the definition (8.12), we compute

∇ 3

ˆ

µ

r3

˙

“ ∇ 3pdiv ζ ` ρ´ 4 pF qρ ˇpF qρ´ 2r pF qρdiv pF qβq

“ ´ pκ´ 2ωq div ζ ` 2D1D‹1pω, 0q ´ˆ

1

2κ` 2ω

˙

div η

`1

2κdiv ξ ´ div β ´ pF qρdiv pF qβ

´3

2κρ´

ˆ

3

2ρ` pF qρ2

˙

pκ´ κΩq ´ 2κ pF qρ ˇpF qρ´ div β ´ pF qρ div pF qβ

`4κ pF qρ ˇpF qρ´ 4 pF qρp´κ ˇpF qρ´ pF qρ`

κ´ κΩ˘

´ div pF qβq

´κr pF qρdiv pF qβ ` 2rκ pF qρdiv pF qβ

´2r pF qρp´ pκ´ 2ωq div pF qβ ´ D1D‹1p ˇpF qρ, ˇpF qσq ` 2 pF qρdiv ηq

239

which gives

∇ 3

ˆ

µ

r3

˙

“ ´3

2κdiv ζ ` 2D1D‹1pω, 0q `

ˆ

1

2κ` 2ω

˙

pdiv ζ ´ div ηq

`1

2κdiv ξ ´ 2div β ` 2 pF qρdiv pF qβ

´3

2κρ´

ˆ

3

2ρ´ 3 pF qρ2

˙

pκ´ κΩq ` 6κ pF qρ ˇpF qρ

`p3κ´ 4ωq r pF qρdiv pF qβ ` 2r pF qρD1D‹1p ˇpF qρ, ˇpF qσq ´ 4r pF qρ2div η

Using again the definition of µ in the above, we obtain (A.7).

240

Appendix B

Proofs of Lemma 5.1.1.1 and

Lemma 5.1.1.4

B.1 Proof of Lemma 5.1.1.1

Recall the general coordinate transformation

u “ u` εg1pu, r, θ, φq

r “ r ` εg2pu, r, θ, φq

θ “ θ ` εg3pu, r, θ, φq

φ “ φ` εg4pu, r, θ, φq

241

We compute the differentials:

du “ du` ε ppg1qudu` pg1qrdr ` pg1qθdθ ` pg1qφdφq

dr “ dr ` ε ppg2qudu` pg2qrdr ` pg2qθdθ ` pg2qφdφq

dθ “ dθ ` ε ppg3qudu` pg3qrdr ` pg3qθdθ ` pg3qφdφq

dφ “ dφ` ε ppg4qudu` pg4qrdr ` pg4qθdθ ` pg4qφdφq

The linear expansion of the differentials give

dudr “ dudr ` ε´

pg2qudu2` pg2qrdudr ` pg2qθdudθ ` pg2qφdudφ` pg1qududr

`pg1qrdr2` pg1qθdrdθ ` pg1qφdrdφ

¯

du2“ du2

` ε`

2pg1qudu2` 2pg1qrdudr ` 2pg1qθdudθ ` 2pg1qφdudφ

˘

dθ2“ dθ2

` ε`

2pg3qududθ ` 2pg3qrdrdθ ` 2pg3qθdθ2` 2pg3qφdθdφ

˘

dφ2“ dφ2

` ε`

2pg4qududφ` 2pg4qrdrdφ` 2pg4qθdθdφ` 2pg4qφdφ2˘

We also linearize the functions

r2“ r2

` ε p2rg2q

Ωprq “ Ωprq ` ε pBrΩ ¨ g2q

sin2 θ “ sin2 θ ` ε p2 sin θ cos θg3q

242

The linear expansion of the metric (3.14) becomes

gM,Q “ ´2dudr ` ε´

´ 2pg2qudu2´ 2pg2qrdudr ´ 2pg2qθdudθ ´ 2pg2qφdudφ

´2pg1qududr ´ 2pg1qrdr2´ 2pg1qθdrdθ ´ 2pg1qφdrdφ

¯

`Ωprqdu2` ε

´

2Ωprqpg1qudu2` 2Ωprqpg1qrdudr ` 2Ωprqpg1qθdudθ

`2Ωprqpg1qφdudφ¯

` ε pBrΩ ¨ g2q pdu2q

`r2pdθ2

` sin2 θdφ2` ε

`

2pg3qududθ ` 2pg3qrdrdθ ` 2pg3qθdθ2` 2pg3qφdθdφ

˘

`ε´

2 sin2 θpg4qududφ` 2 sin2 θpg4qrdrdφ` 2 sin2 θpg4qθdθdφ

`2 sin2 θpg4qφdφ2` 2 sin θ cos θg3dφ

2¯

q

`ε p2rg2q pdθ2` sin2 θdφ2

q

The metric in Bondi gauge is of the form (2.1). In particular the following terms

do not appear in the Bondi form:

dr2, drdθ, drdφ

• The only term in dr2 is:

εp´2pg1qrdr2q “ 0

which then implies

Brpg1q “ 0, g1 “ g1pu, θ, φq (B.1)

243

• The terms in drdθ are

´2pg1qθdrdθ ` 2r2pg3qrdrdθ “ 0

which gives

pg3qr “1

r2pg1qθ

and therefore, using (B.1)

g3 “ ´1

rpg1qθpu, θ, φq ` j3pu, θ, φq (B.2)

for any function j3pu, θ, φq.

• The terms in drdφ are

´2pg1qφdrdφ` 2r2 sin2 θpg4qrdrdφ “ 0

which gives

pg4qr “1

r2 sin2 θpg1qφ

and therefore

g4 “ ´1

r sin2 θpg1qφpu, θ, φq ` j4pu, θ, φq (B.3)

The Bondi form of the metric also imposes ∇ 4pςq “ 0. We first compute ς, which we

244

can read off from the term dudr. The terms with dudr are

´2dudr ` εp´2pg2qr ´ 2pg1qu ` 2Ωprqpg1qrqdudr “ p´2` εp´2pg2qr ´ 2pg1quqqdudr

which gives

ς “ 1` εppg2qr ` pg1quq (B.4)

From (B.4), we obtain

Brς “ Brp1` εppg2qr ` pg1quqq “ εB2rpg2q “ 0

which gives

g2 “ r ¨ w1pu, θ, φq ` w2pu, θ, φq (B.5)

Putting together (B.1), (B.2), (B.3) and (B.5), we obtain the general expression

for coordinate transformations preserving the Bondi metric, and therefore proving

Lemma 5.1.1.1.

We summarize here the derivation of the metric component b (which can be read

off the terms dudθ and dudφ): The terms with dudθ are

´2pg2qθdudθ ` 2Ωprqpg1qθdudθ ` 2r2pg3qududθ

which gives

bθ “ ´ε

r2

`

´2pg2qθ ` 2Ωprqpg1qθ ` 2r2pg3qu

˘

(B.6)

245

The terms with dudφ are

´2pg2qφ ` 2Ωprqpg1qφ ` 2r2 sin2 θpg4qu

which gives

bφ “ ´ε

r2 sin2 θ

`

´2pg2qφ ` 2Ωprqpg1qφ ` 2r2 sin2 θpg4qu˘

(B.7)

B.2 Proof of Lemma 5.1.1.4

We derive here the relation between the coordinate transformation



θA “ θA ` ε`

D‹1pg1, 0qpu, θ, φq ` jApu, θ, φq

˘

and the null frame transformation that brings te4, e3, eAu into te4, e3, eAu, i.e.

e4 “ λ`

e4 ` fAeA

˘

,

e3 “ λ´1`

e3 ` fAeA

˘

,

eA “ OAB eB `

1

2fAe4 `

1

2fAe3

Consider the vectorfield e4 associated to the metric (5.1), i.e. e4 “ Br. Then the

vectorfield e4 associated to the Bondi form obtained after the change of coordinates

246

is given by

e4 “B

Br“Bu

Br

B

Bu`Br

Br

B

Br`Bθ

Br

B

Bθ`Bφ

Br

B

Bφ

“ p1` εw1qB

Br` pεpg3qrq

B

Bθ` pεpg4qrq

B

Bφ

(B.8)

On the null frame transformation side, we write

e4 “ λe4 ` λfAeA

“ λB

Br` λfA

B

BθA

(B.9)

Putting (B.8) and (B.9) to be equal we obtain

λ “ 1` εw1 (B.10)

λf θ “ εpg3qr, λfφ “ εpg4qr (B.11)

Using (B.2) and (B.3) we obtain from the last relation

f θ “ εpg3qr “ ε1

r2pg1qθ (B.12)

fφ “ εpg4qr “ ε1

r2 sin2 θpg1qφ (B.13)

which gives

fA “ ´εD‹1pg1, 0qA (B.14)

Consider the vectorfield e3 associated to the metric (5.1), i.e. e3 “ 2Bu ` ΩprqBr.

Then the vectorfield e3 associated to the Bondi form obtained after the change of

247

coordinates is given by

e3 “ 2ς´1 B

Bu` Ω

B

Br` bA

B

BθA

“ 2ς´1

˜

Bu

Bu

B

Bu`Br

Bu

B

Br`Bθ

Bu

B

Bθ`Bφ

Bu

B

Bφ

¸

`Ω

ˆ

p1` εpg2qrqB

Br` pεpg3qrq

B

Bθ` pεpg4qrq

B

Bφ

˙

`bθ

˜

Bu

Bθ

B

Bu`Br

Bθ

B

Br`Bθ

Bθ

B

Bθ`Bφ

Bθ

B

Bφ

¸

` bφ

˜

Bu

Bφ

B

Bu`Br

Bφ

B

Br`Bθ

Bφ

B

Bθ`Bφ

Bφ

B

Bφ

¸

which gives

e3 “`

2ς´1p1` εpg1quq

˘ B

Bu``

ε2ς´1pg2qu ` Ωp1` εpg2qrq

˘ B

Br

``

ε2ς´1pg3qu ` εΩpg3qr ` b

θ˘ B

Bθ``


φ˘ B

Bφ

where we used that b “ Opεq according to (B.6) and (B.7). The coefficient of B

Bθusing

(B.6) reduces to


θ

“ ε2pg3qu ` εΩprqpg3qr ´ε

r2

`

´2pg2qθ ` 2Ωprqpg1qθ ` 2r2pg3qu

˘

“ εΩprq1

r2pg1qθ ´

ε

r2p´2pg2qθ ` 2Ωprqpg1qθq

“ε

r2p2pg2qθ ´ Ωprqpg1qθq

On the null frame transformation side, we have

e3 “ λ´1e3 ` λ´1fAeA

“ 2λ´1 B

Bu` λ´1Ωprq

B

Br` λ´1fA

B

BθA

(B.15)

248

By putting to be equal the coefficient of B

Bθwe obtain

λ´1f θ “ε


f θ “ λε


“ε


(B.16)

which gives

fA “ ε p2D‹1p´g2, 0q ` ΩprqD‹1pg1, 0qq

Consider the vectorfield eθ associated to the metric (5.1), i.e. eθ “ Bθ. Then the

vectorfield eA associated to the Bondi form obtained after the change of coordinates

is given by

eθ “B

Bθ“Bu

Bθ

B

Bu`Br

Bθ

B

Br`Bθ

Bθ

B

Bθ`Bφ

Bθ

B

Bφ

“ εpg1qθB

Bu` εpg2qθ

B

Br` p1` εpg3qθq

B

Bθ` εpg4qθ

B

Bφ

(B.17)


eθ “ Oθθeθ Òθ

φeφ `1

2fθe4 `

1

2fθe3

“ Oθθ B

BθÒθ

φ B

Bφ`

1

2fθ

B

Br`

1

2fθp2

B

Bu` Ωprq

B

Brq

“ fθB

Bu`

ˆ

1

2fθ`

1

2fθΩprq

˙

B

BrÒθ

θ B

BθÒθ

φ B

Bφ

(B.18)

By putting the coefficient of B

Bθto be equal we obtain

Oθθ“ p1` εpg3qθq

249

By using (B.2), we obtain

pg3qθ “ ´1

rB

2θpg1q ` Bθpj3q

giving

Oθθ“ 1` ε

ˆ

´1

rB

2θpg1q ` Bθpj3q

˙


Bφto be equal we obtain

Oθφ“ εpg4qθ


pg4qθ “ 2cos θ

r sin3 θpg1qφ ´

1

r sin2 θBθBφpg1q ` Bθj4

giving

Oθφ“ ε

ˆ

2cos θ

r sin3 θpg1qφ ´

1

r sin2 θBθBφpg1q ` Bθj4

˙

Consider the vectorfield eφ associated to the metric (3.14), i.e. eφ “ Bφ. Then the

vectorfield eφ associated to the Bondi form obtained after the change of coordinates

is given by

eφ “B

Bφ“Bu

Bφ

B

Bu`Br

Bφ

B

Br`Bθ

Bφ

B

Bθ`Bφ

Bφ

B

Bφ

“ εpg1qφB

Bu` εpg2qφ

B

Br` εpg3qφ

B

Bθ` p1` εpg4qφq

B

Bφ

(B.19)

250


eφ “ Oφθeθ Òφ

φeφ `1

2fφe4 `

1

2fφe3

“ Oφθ B

BθÒφ

φ B

Bφ`

1

2fφ

B

Br`

1

2fφp2

B

Bu` Ωprq

B

Brq

“ fφB

Bu`

ˆ

1

2fφ`

1

2fφΩprq

˙

B

BrÒφ

θ B

BθÒφ

φ B

Bφ

(B.20)


Bθto be equal we obtain

Oφθ“ εpg3qφ


pg3qφ “ ´1

rBφBθpg1q ` Bφpj3q

giving

Oφθ“ ε

ˆ

´1

rBφBθpg1q ` Bφpj3q

˙


Bφto be equal we obtain

Oφφ“ 1` εpg4qφ


pg4qφ “ ´1

r sin2 θB

2φpg1q ` Bφj4

251

giving

Oφφ“ 1` ε

ˆ

´1

r sin2 θB

2φpg1q ` Bφj4

˙

All the conditions above together prove the Lemma.

252

Notations

a: gauge function, as in Lemma 5.1.2.1

a: angular momentum parameter in linearized Kerr-Newman solutions, as in Propo-

sition 5.2.2.1

α, α: curvature component, defined in (1.17)

β, β: curvature component, defined in (1.17)

β, β: gauge-invariant quantity, defined in (5.13)

b: metric component in Bondi form (2.1)

b: magnetic charge parameter in linearized Reissner-Nordstrom solutions, as in Propo-

sition 5.2.1.1

χ, χ: Ricci coefficient, defined in (1.13)

η, η: Ricci coefficient, defined in (1.13)

f, f: gauge-invariant quantity, defined in (5.10)

g: metric component in Bondi form (2.1)

gM,Q: Reissner-Nordstrom metric, defined in (3.5)

h, h: gauge function, as in Lemma 5.1.2.1

κ, κ: Ricci coefficient, defined in (1.14)

K: Gauss curvature

M: Reissner-Nordstrom manifold, defined in (3.1)

M: mass parameter in linearized Reissner-Nordstrom solutions, as in Proposition

253

5.2.1.1

µ: charge-mass aspect function defined in (8.12)

Ω: metric component in Bondi form (2.1)

ω, ω: Ricci coefficient, defined in (1.13)

p: gauge-invariant quantity defined in (6.5)

q1, q2: gauge function, as in Lemma 5.1.3.1

q, qF: gauge-invariant quantity defined in (6.5)

Q: electric charge parameter in linearized Reissner-Nordstrom solutions, as in Propo-

sition 5.2.1.1

r: radial function defined by (2.3)

ρ: curvature component, defined in (1.17)

ξ, ξ: Ricci coefficient, defined in (1.13)

s: coordinate in Bondi form (2.1)

S : solution of the linearized Einstein-Maxwell equations as in Definition 4.2.1

σ: curvature component, defined in (1.17)

ς: metric component in Bondi form (2.1)

u: coordinate in Bondi form (2.1)

Υ: scalar function defined in (3.9)

Ξ: quantity defined in (10.78)

ζ: Ricci coefficient, defined in (1.13)

254

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